Derivatives Lecture Notes

in this lesson we&#39;re going to focus on finding the derivative of a function so let&#39;s start with a constant the derivative of any constant is equal to zero so for instance the derivative of the constant 5 is 0 and the derivative let&#39;s say of negative 7 is also 0 and you might be wondering what exactly is a derivative a derivative is a function that gives you the slope at some x value so let&#39;s say if we have the function f of X is equal to 8 if we were to graph this function it would look like a straight line at y equals 8 so around this region let&#39;s say that&#39;s a ye Queene now what is the slope of a straight line the slope of a straight line is 0 and so if you were to find the derivative of this function represented by F prime of X that will give you 0 if you see D over DX it means that you&#39;re about to differentiate something with respect to X so this means the derivative of f of X is equal to F prime of X now how can we find the derivative of a monomial for example what is the derivative of x squared now there&#39;s something called the power rule and the power rule is very useful for finding the derivative of monomials so here&#39;s the formula that you want to use the derivative of a variable raised to a constant such as X to the N is equal to n times X raised to the N minus 1 and so that&#39;s the form of that you could use to find the derivative of a monomial so in this case n is equal to 2 so the derivative of x squared is going to be 2 X to the 2 minus 1 which is 1 or basically 2 to the X power so that&#39;s the derivative of x squared now let&#39;s try some other examples so using the same formula that is the power rule go ahead and find the derivative of these functions so find the derivative of X cube X to the fourth and also X to the fifth power so the derivative of X cube is going to be 3 in this case Anna story so it&#39;s 3 X raised to the 3 minus 1 and so 3 minus 1 is 2 so the answer is going to be 3 x squared now the derivative of X to the fourth power in this case n is 4 so it&#39;s going to be 4 X raised to the 4 minus 1 and 4 minus 1 is 3 so it&#39;s 4 X cube now for the last one in this case n is 5 so it&#39;s going to be 5 X the raised to the 5 minus 1 and 5 minus 1 is 4 so it&#39;s 5 X to the fourth power and so that&#39;s a simple way in which you could find the derivative of a function now let&#39;s say if you want to find the derivative of 4x to the seventh power how would you do it how would you find the derivative of that particular monomial so what we need to do is use something called the constant multiple rule and here it is the derivative of a constant times a function with say a monomial it&#39;s going to be the constant times the derivative of that monomial or that function let&#39;s just put f of X here so in this case our C value is 4 and f of X is X to the seventh so I&#39;m just gonna color code it so C is 4 as you can see here and f of X is X to the seventh so let&#39;s bring out the 4 to the front and then we&#39;re going to multiply it by the derivative of x to the seventh now using the power rule we can find the derivative of that function so it&#39;s gonna be 7 X raised to the 7 minus 1 and so 7 minus 1 is 6 so we have 7 X to the sixth power and now we can multiply 4 &amp; 7 4 times 7 is 28 so the answer is gonna be 28 X to the 6th power and so this is it now let&#39;s try some more examples so go ahead and find the derivatives of these two monomials 8 X to the 4th and also the derivative of 5 x raised to the sixth power so go ahead and take a minute so what we&#39;re going to do is move the constant to the front and using the power rule we&#39;re going to differentiate x to the 4th so the derivative of x to the 4th is going to be 4 times X raised to the 4 minus 1 and 4 minus 1 is 3 and now we need to multiply 8 by 4 8 times 4 is 32 so the answer is 32 X to the 3rd power now for the next one let&#39;s move the constant to the front and then we&#39;re going to multiply by the derivative of X to the sixth power so we can take this exponent move it to the front so this is going to be 5 times the derivative of X to the 6 which is 6 X to the 6 minus 1 which is 5 six x to the fifth power and now let&#39;s multiply 5 times 6 5 times 6 is 30 so it&#39;s 30 X to the fifth power now for the sake of practice let&#39;s try a few more examples so let&#39;s try the derivative of 9 X to the fifth power and also the derivative of 6 X to the seventh power so go ahead and try those two examples so this is going to be 9 times the derivative of X to the fifth power and the derivative of X to the fifth power is 5 X to the fourth power and so 9 times 5 that&#39;s going to be 45 so the answer is 45 X to the fourth power and so that&#39;s it for this one now for the last one it&#39;s going to be 6 times the derivative of X to the seventh power and the derivative of X to the seventh power is going to be we could take the seven move it to the front so that&#39;s going to be seven X to the seven minus one which is 6 and 6 times 7 is 42 so it&#39;s 42 X raised to the sixth power now we said that the derivative of x squared is equal to 2x now how do we know that by the way let&#39;s say if f of X is x squared that means that the derivative of f of X which is F prime of X is 2x but how can we confirm this now recall the derivative is a function that can give you the slope at some x value so we&#39;re going to show that soon but first is there another way in which we can get this answer besides using the power rule and in a typical calculus course you need to know what that way is and sometimes it&#39;s referred to as the definition of the derivative perhaps you&#39;ve seen this function f prime of X is equal to the limit as H approaches zero of f of X plus h minus f of X divided by H now you might be wondering what exactly is f of X plus h what is that well we know what f of X is f of X is x squared so if f of X is x squared what is f of X plus h all you need to do is replace the X with X plus h so inside here we had x squared so instead of X we&#39;re gonna replace it with X plus h so f of X plus h is X plus h square now let&#39;s plug it into this formula so let me erase this so remember we&#39;re trying to show that F prime of X is equal to 2x I&#39;m gonna erase that soon right now we have this F prime of X is equal to the limit as H approaches 0 and this right here we know it&#39;s X plus h squared minus f of X which is x squared all divided by H now what do you think we need to do at this point what&#39;s our next step our next step is to foil that expression so this is equivalent to the limit as H approaches 0 and X plus h squared is the same as X plus h times X plus h so let&#39;s go ahead and foil that expression now when taken a calculus exam you will need to rewrite the limit expression even though it might be tedious some teachers will actually take off points if you don&#39;t rewrite it so here we have x times X and that&#39;s gonna be x squared and then we have x times H and then it&#39;s H times X which is the same as x times H and then the last one H times H so that&#39;s H squared and then minus x squared divided by H now at this point we can cancel the x squared term and we can combine like terms XH plus XH that&#39;s 2 XH so now we have 2x h plus h squared divided by H now our next step is to factor the GCF that is the greatest common factor which is H so if we take out an H from 2 X H that&#39;s gonna be 2 X + H squared divided by H will give us H so now we can cancel H so now what we have left over is this the limit as H approaches 0 of 2 X + H and so when H becomes 0 this is going to be 2x + 0 so basically H disappears as H approaches 0 and that&#39;s how we get the final answer 2x and so that&#39;s why derivative of x squared is 2x and so that&#39;s how you can find this answer using the limit process now we said that the derivative is a function we&#39;ll give you the slope at any x value so let&#39;s say that f of X is x squared and we wish to find the slope of the tangent line at x equals 1 so we know what F prime of X is using the power rule it&#39;s 2x and so to find the slope at x equals 1 we need to evaluate F prime of X when X is 1 and so that&#39;s gonna be 2 times 1 which is 2 so the slope of the tangent line should be equal to 2 now if you were to draw a rough sketch of the graph y equals x squared it will look something like this and when x is equal to 1 the slope of the tangent line will equal 2 so I&#39;m gonna put Mt or M 10 so the slope of the tangent line is 2 when X is equal to 1 and so that&#39;s what the derivative function tells you it gives you the slope of the tangent line at some exit value now you need to know the difference between a tangent line and a secant line a secant line is basically a line that touches the curve at two points and I missed it so let&#39;s do that again and the tangent line is a line that touches the curve only at one point so make sure you know the difference between the two now in algebra you&#39;ve learned that to find the slope of a line you need two points and this is basically finding the slope of a secant line that&#39;s on a curve so let&#39;s put M secant and you know that as y2 minus y1 equals x2 minus x1 so we could take two points on this curve and basically get a secant line and as those two points approach at this point the slope of the secant line approaches the slope of the tangent line now we need to pick two points where the midpoint of those two points is x equals 1 so we can choose let&#39;s say 0.9 and 1.1 as our x1 and x2 values because if you add up those two numbers and divide by 2 the average of 0.9 and 1.1 is 1 or we could pick point in the high 9 and 1 point 0 1 because the midpoint of those two numbers is still 1 however 0.99 and 1 point 0 1 is closer to 1 than point 9 and 1 point 1 so the slope of the secant line based on these two values will be a lot closer to the slope of the tangent line at x equals 1 and so let&#39;s go ahead and calculate those values so let&#39;s say that x1 is 0.9 to begin with and x2 is 1 point 1 and let&#39;s use this formula to calculate the slope of the secant line now keep in mind the slope of the tangent line is this number it&#39;s equal to 2 so this is going to be Y to the minus y1 divided by x2 minus x1 and so y2 corresponds to the Y value for this x value and Y is equal to f of X so we could use this function f of x equals x squared to find y2 so when X 2 is 1 point 1 y2 is 1 point 1 squared because y equals x squared when X 1 is 0.9 and y1 is 0.9 squared now one point 1 squared that&#39;s one point 2 1 and point 9 squared is 0.8 e1 and 1 point 1 the minus 0.9 is point 2 1.21 - point eight one divided by point two gives us already an exact answer which is two and so there&#39;s no need to use point nine nine in this instance we can see that it&#39;s exactly the same so let&#39;s try an example where it may not be exactly the same so this time let&#39;s say that f of X is x cubed and we wish to calculate the slope of this tangent line at x equals two so we know what F prime of X is the derivative of x cubed using the power rule is three x squared so the derivative at x equals two is going to be 3 times 2 squared 2 squared is 2 times 2 that&#39;s 4 times 3 is 12 so the slope of the tangent line at x equals 2 is 12 now let&#39;s see if we can approximate this value with the slope of the secant line so let&#39;s choose an X 1 value of 1.9 and an X 2 value of 2.1 and so the slope of the secant line between those two points is going to be y2 minus y1 over x2 minus x1 so in this case we said X 2 is 2.1 now what&#39;s y2 y2 has to be 2 point 1 raised to the third power because Y is equal to X cubed x 1 is 1 point 9 so y1 is 1 point 9 to the third power now 2 point 1 raised to the third power that&#39;s gonna be nine point two six one and one point nine raised to the third power that&#39;s going to be six point eight five nine and two point one minus one point nine that&#39;s point two nine point two six one minus six point eight five nine that&#39;s two point four zero two and if we divide that by point two it gives us a very good approximation actually twelve point zero one and so you could see that you can approximate the slope of this tangent line using the slope of the secant line and that&#39;s what the derivative tells you it gives you the slope of the tangent line which touches the curve at one point that&#39;s some x value interview remember this the derivative is a function and that helps you to find the slope of a tangent line at some a value of x so keep that in mind now let&#39;s talk about finding the derivative of a polynomial function so let&#39;s say that f of X is x cubed plus 7x squared minus 8x plus 6 what is the derivative of that function so what is f prime of X so go ahead and work on this problem so what we need to do is differentiate each monomial separately using the power rule the derivative of x cubed is 3x squared now what about the derivative of 7x squared using a constant multiple rule it&#39;s going to be seven times the derivative of x squared which is 2x or 2 X to the first power and seven times 2x is 14x now what about the derivative of negative 8x what is that equal to so keep in mind this is negative 8 times X to the first power so this is going to be negative 8 times the derivative of x to the first power now what is the derivative of X to the first power well using the power rule you need to move the 1 to the front so it&#39;s 1 times X raised to the 1 minus 1 which becomes negative 8 times 1 exit is 0 now what is X is 0 exit is 0 or anything raised to the 0 power is 1 so this becomes negative 8 times 1 which is just negative 8 so the derivative of negative 8x is simply negative 8 and the derivative of any constant is 0 so we could stop it here this is the answer f prime of X is 3x squared plus 14x and minus 8 the video that you&#39;re currently watching is the first part of the entire video for those of you who want access to the second part of the video I&#39;m gonna put it on my patreon page as you can see the link in the screen and on that page I have some other video content that you might be interested in so feel free to take a look at that when you get a chance now let&#39;s get back to the lesson now let&#39;s say that f of X is 4 X to the fifth power plus 3x to the fourth power plus 9 X minus 7 what is f prime of X so go ahead and try this for the sake of practice so using the constant multiple rule we&#39;re going to rewrite the constant and take the derivative of X to the fifth power using the power rule and so that&#39;s going to be 5x to the fourth power now the derivative of X to the fourth is 4x cubed and the derivative of X is always just 1 and the derivative of a constant is zero and so we have 4 times 5 which is 20 and 3 times 4 which is 12 and so this is the final answer f prime of X is 20 X to the fourth power plus 12x cubed plus 9 now let&#39;s say that f of X is 2x to the fifth power plus let&#39;s say 5x to the third power plus 3x squared plus 4 and so you&#39;re given this function and you&#39;re told to find the slope of the tangent line at x equals 2 go ahead and try this problem anytime we need to find a slope with the tangent line first you need to find the derivative of the function that is f prime of X and then simply plug in the x value into that function so let&#39;s determine F prime of X versus the derivative of X to the fifth power is 5x to the fourth power and the derivative of x cubed is 3x squared and the derivative of x squared is 2x and for the constant we just don&#39;t need to worry about it so f prime of X is going to be 10 2 times 5 is 10 so 10 X to the fourth power 5 times 3 is 15 3 times 2 is 6 and so this is what we have now to calculate the slope and let&#39;s replace X with Thun now Q to the fourth power if we multiply to 4 times 2 times 2 is 4 times 2 is 8 times 2 is 16 and then 2 squared that&#39;s 4 and 6 times 2 is 12 so now we have F prime of two is equal to ten times 16 which is 160 15 times four that&#39;s 60 and 160 plus 60 that&#39;s 220 and so the final answer is going to be 232 so that&#39;s the slope of the tangent line when X is equal to two now let&#39;s say that f of X is 1 over X what is the derivative of that function so what is the derivative of 1 over X how would you go about fighting it for a situation like this you need to rewrite the function and so we need to do is take the X variable and move it to the top when you do that the exponent changes signed it&#39;s gonna change from positive 1 it&#39;s a negative 1 now at this point you could use the power rule so remember the derivative of X to the N is n X raised to the N minus 1 so in this case n is negative 1 and negative 1 minus 1 is negative 2 so we have negative 1 X to the negative 2 now once you have the derivative you need to rewrite it into a more proper form so let&#39;s take the X variable and move it back to the bottom so our final answer will look like this it&#39;s negative 1 divided by x squared and so that&#39;s how you could find the derivative of a rational function now let&#39;s say that f of X is 1 over x squared go ahead and find f prime of x for the sake of practice so take a minute and try that example just like before we&#39;re going to rewrite the function so let&#39;s move the X variable to the numerator of the fraction so f of X is equivalent to X raised to the negative 2 now at this point after you rewrite it now you can find the derivative using the so let&#39;s take the constant I mean of the constant but the exponent move it to the front so our n value is negative two and then let&#39;s subtract negative 2 by one negative two minus one is negative three and now let&#39;s rewrite this expression by taking the X variable and moving it to the bottom if you&#39;re wondering why I divided by one it&#39;s the same thing negative two X to the negative 3 is the same as negative two X to the negative three over one I just like to write it in a fraction so you could see what I&#39;m going to do next and that is moving the X variable to the bottom so now the exponent will change from negative three it&#39;s a positive three and so this is the final answer so this is the derivative of 1 over x squared now let&#39;s try another example so let&#39;s say that f of X let&#39;s try a harder example let&#39;s say it&#39;s 8 over X to the fourth power go ahead and work on that problem so let&#39;s rewrite the function let&#39;s move the X variable to the top and so this is gonna be 8 X raised to the negative fourth power and now let&#39;s differentiate the function so we need to use the constant multiple rule in this case so it&#39;s going to be eight times the derivative so I&#39;m going to write that as times D over DX the derivative of x to the negative fourth power and so in this case using the power rule our and the value is negative 4 so it&#39;s gonna be times negative 4x raised to the negative 4 minus 1 and so negative 4 minus 1 is negative 5 and 8 times negative 4 is negative 32 so the answer is negative 32 to the negative 5th power well let&#39;s rewrite it though let&#39;s not leave it like that so if we move the X to the bottom we could write the final answer fully simplified as a negative 32 divided by X to the fifth power now let&#39;s talk about finding the derivative of radical functions for instance what is the derivative of the square root of x so what do you think we need to do now the first thing we need to do is rewrite this expression as a rational exponent so X is the same as X to the first power and if you don&#39;t see an index number it&#39;s always a 2 so this is equivalent to X raised to the 1/2 now in this form we can use the power rule so n is going to be 1/2 so it&#39;s 1/2 X raised to the 1/2 minus 1 now we need to subtract a fraction by a whole number and so we need to get common denominators 1 is the same as 2 divided by 2 2 divided by 2 is 1 and 1 over to the -2 over 2 that&#39;s gonna be negative 1 over 2 not because we have a negative exponent we need to rewrite this so right now the 1 is on top and the X is also on top the 2 is in the bottom of the fraction but now I need to take the X variable and move it to the bottom so this becomes 1 over 2 times X raised to the positive 1/2 and at this point I can rewrite the rational exponent as a radical so we know that X to the 1/2 is the square root of x so the final answer is 1 divided by 2 square root X and so this is it now let&#39;s work on another example so let&#39;s say that f of X is the cube root of x to the fifth power so what is f prime of X go ahead and try that so let&#39;s begin by rewriting this expression so the cube root of x to the fifth power we can rewrite that as a rational exponent and it&#39;s going to look like this it&#39;s x raised to the 5 over 3 so this number here becomes the numerator of the rational exponent and the index number becomes the denominator of the fraction that we see here now let&#39;s use the power rule so in this case n is a fraction it&#39;s going to be 5 over 3 and then we&#39;ll have X raised to the N minus 1 so that&#39;s 5 over 3 minus 1 now just like before we need to get common denominators 1 is the same as 3 divided by 3 and so we have 5 over 3 the minus 3 over 3 5 minus 3 is 2 so that becomes 2 over 3 and I&#39;m going to rewrite this as 5 times X raised to 2/3 divided by 3 now because the exponent is still positive we don&#39;t need to move the X variable to the bottom and that&#39;s not necessary the last thing that we need to do is convert this back into a radical expression so this is going to be 5 times the cube root of x squared over 3 and so this is the final answer here&#39;s another one that you can work on so let&#39;s say that we have the monomial or rather just a radical expression the 7th root of x to the fourth power what is the derivative of that expression so this is X raised to the 4 over 7 and let&#39;s use the power rule so n is 4 over 7 and it&#39;s gonna be X to the 4 over 7 minus 1 now to get common denominators let&#39;s replace one with 7 over 7 now 4 over 7 minus 7 over 7 that&#39;s a negative 3 over 7 so I&#39;m gonna rewrite this as a fraction the fours on top the X variable is currently on top but the 7 is in the bottom of the fraction now in this case we do have a negative exponent so we need to move the X variable to the bottom and so it&#39;s gonna be 4/7 times X raised to the 3 over 7 and now we can rewrite the rational exponent as a radical expression so the final answer is gonna be 4/7 times the seventh root of x cube and that&#39;s it so this is the answer now of course if you want to you can rationalize the denominator but I&#39;m not gonna worry about that in this video now let&#39;s talk about some other problems that you might see in your homework so let&#39;s say if you&#39;re given a problem that looks like this it has x squared on the outside and then within a parenthesis it has X cubed plus 7 how would you find the derivative of this expression what would you do in this case the best thing to do right now with what you already know is to distribute the x squared 2x cubed plus 5 and then you can find the derivative so x squared times X cubed that&#39;s going to be X to the fifth power because 2 plus 3 is 5 and then x squared times 7 is 7x squared so in this form it&#39;s very easy to find the first derivative so the derivative of X to the fifth is 5x to the fourth and the derivative of x squared is 2x so the final answer is 5x to the fourth power plus 14x and so that&#39;s what you need to do if you ever come across like a situation like that now let&#39;s try a different example let&#39;s say that f of X is equal to 2x minus 3 raised to the second power what would you do in this case now there&#39;s something called the chain rule which we can use here but you haven&#39;t learned that yet so we&#39;ll save that for another day or rather later on in this video something we can do is expand this expression so whenever you see an exponent of 2 whatever that exponent is attached to it means that you have two of these things and multiplied to each other so this expression is equivalent to 2x minus 3 times another 2x minus 3 which that does not look like a 3 and so what we&#39;re doing is we&#39;re multiplying a binomial by another binomial and so let&#39;s use the foil method so let&#39;s multiply the first two terms 2x times 2x 2 times 2 is 4 x times X is x squared and then 2x times negative 3 that&#39;s negative 6x negative 3 times 2x is also negative 6x and then we have negative 3 times negative 3 which is positive 9 now let&#39;s combine like terms so negative 6x plus negative 6x is the negative 12x and now we could find the first derivative the derivative of x squared is 2x the derivative of x is 1 and for a constant is 0 so the final answer is going to be 8x minus 12 and so that&#39;s the derivative of 2x and minus 3 we&#39;re so that&#39;s what you could do in a situation like this now let&#39;s say we have a fraction X to the fifth plus 6x to the fourth power plus 5x cube divided by x squared in this case what is the derivative of f of X now based on the previous examples you know you need to simplify this before finding the derivative so how can you simplify this expression if you&#39;re dividing a trinomial by a monomial what you could do is divide every term by x squared separately so let&#39;s begin by dividing X to the fifth by x squared and it&#39;s a point to understand that when you multiply let&#39;s say x squared by X cube we need to add the exponents when you divide you need to subtract so this is 5 minus 2 that&#39;s X cubed and so that&#39;s going to be the first part so X to the fifth power divided by x squared is X cubed now 6 X to the fourth power divided by x squared that&#39;s going to be 6 x squared all you need to do is subtract to the exponents 4 minus 2 is 2 and 5x cubed divided by x squared is going to be 5x to the first power because 3 minus 2 is 1 and so now we have this simplified polynomial and now we can find the first derivative so it&#39;s going to be 3x squared and the derivative of x squared is 2x and the derivative of X is 1 so the final answer is 3x squared plus 12x plus 5 and that concludes this example now let&#39;s talk about the derivatives of trigonometric functions and I&#39;m going to give you a few that you need to know and for now write these downs because we&#39;re going to use this later now the derivative of sine X you need to know is cosine X and the derivative of cosine X is negative sine X so that&#39;s the first two you need to know next you need to know that the derivative of secant X is secant X tangent X and the derivative of cosecant X is actually very similar it&#39;s going to be negative cosecant X cotangent X something that helps me to remember these things is that if you see a C in front you&#39;re going to have a negative sign like the derivative of cosine it was negative sine now consider the last two the derivative of tangent is secant squared and based on that what do you think the derivative of cotangent X will be well notice that we do have the C it turns out that it&#39;s negative cosecant squared X so keep those six derivative functions in mind because we&#39;re gonna be using them later now the next thing that we&#39;re going to go over is the product rule and here it is so let&#39;s say if you have two functions multiplied to each other and you wish to find the derivative of that result it&#39;s going to be the derivative of the first function times the second plus the first function times the derivative of the second so let&#39;s say if we wish to determine the derivative of X times sine X so in this case we could say that f is x squared and G is sine X so I&#39;m going to write it out so if f is x squared what is f prime F prime is the derivative of F and the derivative of x squared is 2x now G is going to be equivalent to sine X and G prime is the derivative of sine X which we now know is cosine X so now at this point all we need to do is basically plug in what we have on the right side of the equation so f prime is 2x G is sine X f is x squared and G prime is cosine X so this is the answer if you want to you can factor out the GCF which is X but I&#39;m going to leave the answer like this and so that&#39;s how you could use the product rule when finding the derivative of functions that are multiplied to each other now let&#39;s try some other examples try this problem what is the derivative of let&#39;s say 3x to the fourth power plus 7 times X cubed minus 5x now granted we can foil this expression because we did an example like that earlier but let&#39;s use the product rule to get the answer feel free to pause the video if you want to so what I&#39;m gonna do first is I&#39;m gonna write the formula so the derivative of F times G is going to be the derivative of the first part times the second plus the first part let me write that again times the derivative of the second so what&#39;s F and what&#39;s G we&#39;re going to say that F is the first part and so we&#39;re going to say that f is 3x to the fourth power plus seven so what is f prime so the derivative of X to the fourth is 4x cubed but we&#39;re going to multiply that by three and so three times four that&#39;s going to give us 12 so this is going to be 12x cubed and the derivative of seven is zero now G is the second part of the function so G is X cubed minus 5x G prime is going to be three x squared and the derivative of X is one times negative five so that&#39;s just going to be negative five and so that&#39;s G prime so using the formula this expression becomes equal to which I&#39;m going to write over here this is going to be F prime which is 12x cubed times G which is X cubed minus 5x plus F and that&#39;s three X to the fourth plus seven times G prime 3x squared minus five and that&#39;s it right there now here is a challenge problem for you what is the derivative of x cubed times tangent x times three x squared minus nine so this time we have three parts being multiplied to each other so we saw how to use the product rule when having two different functions being multiplied to each other but what about three different functions so if the derivative of let&#39;s say a two-part function like F times G if that&#39;s F Prime and G Plus F G Prime what would the derivative of let&#39;s say F times G times H be so this is going to be we&#39;re going to differentiate the first part and then leave the second two parts the same plus we&#39;re going to leave the first part this differentiate the second part and then leave the third part the same and then it&#39;s going to be the first two parts times the derivative of the last part so when using the product rule when you differentiate one part the other two parts should remain the same and then you just go in order differentiate the first part and then the second part and then the third part so once you understand the format or the procedure of doing this you can just go ahead and get the answer without actually writing down what&#39;s F of G and H so first let&#39;s find the derivative of the first part the derivative of x cube is 3x squared now the other two parts the G and H we&#39;re just going to rewrite it for now so it&#39;s going to be times tangent X and then times 3x squared minus 9 now let&#39;s rewrite the first part which is X cube and then we&#39;re going to take the derivative of the second part the derivative of tangent if you remember is secant squared now let&#39;s rewrite the third part which is 3x squared minus 9 now for the last part we&#39;re going to rewrite the first two parts X cube and tangent X but this time we are going to take the derivative of 3x squared minus 9 which is going to be just 6x because this will go to 0 now let&#39;s say if we want to find the derivative of a fraction such as let&#39;s say 5x plus 6 divided by 3x minus 7 in this case you want to use something called the quotient rule and here&#39;s the formula that we&#39;re going to use so the derivative of let&#39;s say F divided by G this is going to be G F - f G prime divided by G squared and this is something that you simply need to commit to memory you just got to know that function at least it works for me when I was in high school I just memorized that function now F is going to be the top portion of this function so in this case F is going to be 5x + 6 F prime is the derivative of F so the derivative of 5x + 6 is 5 now G G is going to be the bottom part of this function so G is going to be 3x - 7 which means G Prime history so if it helps to write everything out by all means go ahead and do that if it makes your life easier or if it helps you to avoid mistakes and on a test one of the biggest things that you have to do is avoid mistakes because if you make a mistake even if you know it I mean that&#39;s just gonna ruin your test score now let&#39;s go ahead and finish this G is 3x minus 7 and then F prime that&#39;s 5 and then we have F which is 5x plus 6 and then G prime that&#39;s 3 and then divided by G squared so G is 3x minus 7 and then let&#39;s square that now in this case I&#39;m going to simplify because it doesn&#39;t require that much work to do so so let&#39;s begin by distributing the 5 to 3x - 7 so 5 times 3x that&#39;s 15 X and then 5 times 7 or 5 times negative 7 rather that&#39;s negative 35 and here we have negative 5x times 3 which is going to be negative 15 x + 6 times 3 is 18 but we got the negative sign so that&#39;s going to be negative 18 and I&#39;m in the foyer the stuff on the bottom because it looks better this way now we could cancel 15 X negative 35 minus 18 that&#39;s going to be negative 53 and so the answer is negative 53 divided by 3x minus 7 squared if it&#39;s easy to simplify your answer feel free but sometimes if it takes a lot of work to simplify it most teachers will allow you just to write the answer the way it is some teachers will allow you to leave the answer like this so you need to basically know your teacher and how they want you to write the final answer

in this lesson we&amp;#39;re going to focus on finding the derivative of a function so let&amp;#39;s start with a constant the derivative of any constant is equal to zero so for instance the derivative of the constant 5 is 0 and the derivative let&amp;#39;s say of negative 7 is also 0 and you might be wondering what exactly is a derivative a derivative is a function that gives you the slope at some x value so let&amp;#39;s say if we have the function f of X is equal to 8 if we were to graph this function it would look like a straight line at y equals 8 so around this region let&amp;#39;s say that&amp;#39;s a ye Queene now what is the slope of a straight line the slope of a straight line is 0 and so if you were to find the derivative of this function represented by F prime of X that will give you 0 if you see D over DX it means that you&amp;#39;re about to differentiate something with respect to X so this means the derivative of f of X is equal to F prime of X now how can we find the derivative of a monomial for example what is the derivative of x squared now there&amp;#39;s something called the power rule and the power rule is very useful for finding the derivative of monomials so here&amp;#39;s the formula that you want to use the derivative of a variable raised to a constant such as X to the N is equal to n times X raised to the N minus 1 and so that&amp;#39;s the form of that you could use to find the derivative of a monomial so in this case n is equal to 2 so the derivative of x squared is going to be 2 X to the 2 minus 1 which is 1 or basically 2 to the X power so that&amp;#39;s the derivative of x squared now let&amp;#39;s try some other examples so using the same formula that is the power rule go ahead and find the derivative of these functions so find the derivative of X cube X to the fourth and also X to the fifth power so the derivative of X cube is going to be 3 in this case Anna story so it&amp;#39;s 3 X raised to the 3 minus 1 and so 3 minus 1 is 2 so the answer is going to be 3 x squared now the derivative of X to the fourth power in this case n is 4 so it&amp;#39;s going to be 4 X raised to the 4 minus 1 and 4 minus 1 is 3 so it&amp;#39;s 4 X cube now for the last one in this case n is 5 so it&amp;#39;s going to be 5 X the raised to the 5 minus 1 and 5 minus 1 is 4 so it&amp;#39;s 5 X to the fourth power and so that&amp;#39;s a simple way in which you could find the derivative of a function now let&amp;#39;s say if you want to find the derivative of 4x to the seventh power how would you do it how would you find the derivative of that particular monomial so what we need to do is use something called the constant multiple rule and here it is the derivative of a constant times a function with say a monomial it&amp;#39;s going to be the constant times the derivative of that monomial or that function let&amp;#39;s just put f of X here so in this case our C value is 4 and f of X is X to the seventh so I&amp;#39;m just gonna color code it so C is 4 as you can see here and f of X is X to the seventh so let&amp;#39;s bring out the 4 to the front and then we&amp;#39;re going to multiply it by the derivative of x to the seventh now using the power rule we can find the derivative of that function so it&amp;#39;s gonna be 7 X raised to the 7 minus 1 and so 7 minus 1 is 6 so we have 7 X to the sixth power and now we can multiply 4 &amp;amp; 7 4 times 7 is 28 so the answer is gonna be 28 X to the 6th power and so this is it now let&amp;#39;s try some more examples so go ahead and find the derivatives of these two monomials 8 X to the 4th and also the derivative of 5 x raised to the sixth power so go ahead and take a minute so what we&amp;#39;re going to do is move the constant to the front and using the power rule we&amp;#39;re going to differentiate x to the 4th so the derivative of x to the 4th is going to be 4 times X raised to the 4 minus 1 and 4 minus 1 is 3 and now we need to multiply 8 by 4 8 times 4 is 32 so the answer is 32 X to the 3rd power now for the next one let&amp;#39;s move the constant to the front and then we&amp;#39;re going to multiply by the derivative of X to the sixth power so we can take this exponent move it to the front so this is going to be 5 times the derivative of X to the 6 which is 6 X to the 6 minus 1 which is 5 six x to the fifth power and now let&amp;#39;s multiply 5 times 6 5 times 6 is 30 so it&amp;#39;s 30 X to the fifth power now for the sake of practice let&amp;#39;s try a few more examples so let&amp;#39;s try the derivative of 9 X to the fifth power and also the derivative of 6 X to the seventh power so go ahead and try those two examples so this is going to be 9 times the derivative of X to the fifth power and the derivative of X to the fifth power is 5 X to the fourth power and so 9 times 5 that&amp;#39;s going to be 45 so the answer is 45 X to the fourth power and so that&amp;#39;s it for this one now for the last one it&amp;#39;s going to be 6 times the derivative of X to the seventh power and the derivative of X to the seventh power is going to be we could take the seven move it to the front so that&amp;#39;s going to be seven X to the seven minus one which is 6 and 6 times 7 is 42 so it&amp;#39;s 42 X raised to the sixth power now we said that the derivative of x squared is equal to 2x now how do we know that by the way let&amp;#39;s say if f of X is x squared that means that the derivative of f of X which is F prime of X is 2x but how can we confirm this now recall the derivative is a function that can give you the slope at some x value so we&amp;#39;re going to show that soon but first is there another way in which we can get this answer besides using the power rule and in a typical calculus course you need to know what that way is and sometimes it&amp;#39;s referred to as the definition of the derivative perhaps you&amp;#39;ve seen this function f prime of X is equal to the limit as H approaches zero of f of X plus h minus f of X divided by H now you might be wondering what exactly is f of X plus h what is that well we know what f of X is f of X is x squared so if f of X is x squared what is f of X plus h all you need to do is replace the X with X plus h so inside here we had x squared so instead of X we&amp;#39;re gonna replace it with X plus h so f of X plus h is X plus h square now let&amp;#39;s plug it into this formula so let me erase this so remember we&amp;#39;re trying to show that F prime of X is equal to 2x I&amp;#39;m gonna erase that soon right now we have this F prime of X is equal to the limit as H approaches 0 and this right here we know it&amp;#39;s X plus h squared minus f of X which is x squared all divided by H now what do you think we need to do at this point what&amp;#39;s our next step our next step is to foil that expression so this is equivalent to the limit as H approaches 0 and X plus h squared is the same as X plus h times X plus h so let&amp;#39;s go ahead and foil that expression now when taken a calculus exam you will need to rewrite the limit expression even though it might be tedious some teachers will actually take off points if you don&amp;#39;t rewrite it so here we have x times X and that&amp;#39;s gonna be x squared and then we have x times H and then it&amp;#39;s H times X which is the same as x times H and then the last one H times H so that&amp;#39;s H squared and then minus x squared divided by H now at this point we can cancel the x squared term and we can combine like terms XH plus XH that&amp;#39;s 2 XH so now we have 2x h plus h squared divided by H now our next step is to factor the GCF that is the greatest common factor which is H so if we take out an H from 2 X H that&amp;#39;s gonna be 2 X + H squared divided by H will give us H so now we can cancel H so now what we have left over is this the limit as H approaches 0 of 2 X + H and so when H becomes 0 this is going to be 2x + 0 so basically H disappears as H approaches 0 and that&amp;#39;s how we get the final answer 2x and so that&amp;#39;s why derivative of x squared is 2x and so that&amp;#39;s how you can find this answer using the limit process now we said that the derivative is a function we&amp;#39;ll give you the slope at any x value so let&amp;#39;s say that f of X is x squared and we wish to find the slope of the tangent line at x equals 1 so we know what F prime of X is using the power rule it&amp;#39;s 2x and so to find the slope at x equals 1 we need to evaluate F prime of X when X is 1 and so that&amp;#39;s gonna be 2 times 1 which is 2 so the slope of the tangent line should be equal to 2 now if you were to draw a rough sketch of the graph y equals x squared it will look something like this and when x is equal to 1 the slope of the tangent line will equal 2 so I&amp;#39;m gonna put Mt or M 10 so the slope of the tangent line is 2 when X is equal to 1 and so that&amp;#39;s what the derivative function tells you it gives you the slope of the tangent line at some exit value now you need to know the difference between a tangent line and a secant line a secant line is basically a line that touches the curve at two points and I missed it so let&amp;#39;s do that again and the tangent line is a line that touches the curve only at one point so make sure you know the difference between the two now in algebra you&amp;#39;ve learned that to find the slope of a line you need two points and this is basically finding the slope of a secant line that&amp;#39;s on a curve so let&amp;#39;s put M secant and you know that as y2 minus y1 equals x2 minus x1 so we could take two points on this curve and basically get a secant line and as those two points approach at this point the slope of the secant line approaches the slope of the tangent line now we need to pick two points where the midpoint of those two points is x equals 1 so we can choose let&amp;#39;s say 0.9 and 1.1 as our x1 and x2 values because if you add up those two numbers and divide by 2 the average of 0.9 and 1.1 is 1 or we could pick point in the high 9 and 1 point 0 1 because the midpoint of those two numbers is still 1 however 0.99 and 1 point 0 1 is closer to 1 than point 9 and 1 point 1 so the slope of the secant line based on these two values will be a lot closer to the slope of the tangent line at x equals 1 and so let&amp;#39;s go ahead and calculate those values so let&amp;#39;s say that x1 is 0.9 to begin with and x2 is 1 point 1 and let&amp;#39;s use this formula to calculate the slope of the secant line now keep in mind the slope of the tangent line is this number it&amp;#39;s equal to 2 so this is going to be Y to the minus y1 divided by x2 minus x1 and so y2 corresponds to the Y value for this x value and Y is equal to f of X so we could use this function f of x equals x squared to find y2 so when X 2 is 1 point 1 y2 is 1 point 1 squared because y equals x squared when X 1 is 0.9 and y1 is 0.9 squared now one point 1 squared that&amp;#39;s one point 2 1 and point 9 squared is 0.8 e1 and 1 point 1 the minus 0.9 is point 2 1.21 - point eight one divided by point two gives us already an exact answer which is two and so there&amp;#39;s no need to use point nine nine in this instance we can see that it&amp;#39;s exactly the same so let&amp;#39;s try an example where it may not be exactly the same so this time let&amp;#39;s say that f of X is x cubed and we wish to calculate the slope of this tangent line at x equals two so we know what F prime of X is the derivative of x cubed using the power rule is three x squared so the derivative at x equals two is going to be 3 times 2 squared 2 squared is 2 times 2 that&amp;#39;s 4 times 3 is 12 so the slope of the tangent line at x equals 2 is 12 now let&amp;#39;s see if we can approximate this value with the slope of the secant line so let&amp;#39;s choose an X 1 value of 1.9 and an X 2 value of 2.1 and so the slope of the secant line between those two points is going to be y2 minus y1 over x2 minus x1 so in this case we said X 2 is 2.1 now what&amp;#39;s y2 y2 has to be 2 point 1 raised to the third power because Y is equal to X cubed x 1 is 1 point 9 so y1 is 1 point 9 to the third power now 2 point 1 raised to the third power that&amp;#39;s gonna be nine point two six one and one point nine raised to the third power that&amp;#39;s going to be six point eight five nine and two point one minus one point nine that&amp;#39;s point two nine point two six one minus six point eight five nine that&amp;#39;s two point four zero two and if we divide that by point two it gives us a very good approximation actually twelve point zero one and so you could see that you can approximate the slope of this tangent line using the slope of the secant line and that&amp;#39;s what the derivative tells you it gives you the slope of the tangent line which touches the curve at one point that&amp;#39;s some x value interview remember this the derivative is a function and that helps you to find the slope of a tangent line at some a value of x so keep that in mind now let&amp;#39;s talk about finding the derivative of a polynomial function so let&amp;#39;s say that f of X is x cubed plus 7x squared minus 8x plus 6 what is the derivative of that function so what is f prime of X so go ahead and work on this problem so what we need to do is differentiate each monomial separately using the power rule the derivative of x cubed is 3x squared now what about the derivative of 7x squared using a constant multiple rule it&amp;#39;s going to be seven times the derivative of x squared which is 2x or 2 X to the first power and seven times 2x is 14x now what about the derivative of negative 8x what is that equal to so keep in mind this is negative 8 times X to the first power so this is going to be negative 8 times the derivative of x to the first power now what is the derivative of X to the first power well using the power rule you need to move the 1 to the front so it&amp;#39;s 1 times X raised to the 1 minus 1 which becomes negative 8 times 1 exit is 0 now what is X is 0 exit is 0 or anything raised to the 0 power is 1 so this becomes negative 8 times 1 which is just negative 8 so the derivative of negative 8x is simply negative 8 and the derivative of any constant is 0 so we could stop it here this is the answer f prime of X is 3x squared plus 14x and minus 8 the video that you&amp;#39;re currently watching is the first part of the entire video for those of you who want access to the second part of the video I&amp;#39;m gonna put it on my patreon page as you can see the link in the screen and on that page I have some other video content that you might be interested in so feel free to take a look at that when you get a chance now let&amp;#39;s get back to the lesson now let&amp;#39;s say that f of X is 4 X to the fifth power plus 3x to the fourth power plus 9 X minus 7 what is f prime of X so go ahead and try this for the sake of practice so using the constant multiple rule we&amp;#39;re going to rewrite the constant and take the derivative of X to the fifth power using the power rule and so that&amp;#39;s going to be 5x to the fourth power now the derivative of X to the fourth is 4x cubed and the derivative of X is always just 1 and the derivative of a constant is zero and so we have 4 times 5 which is 20 and 3 times 4 which is 12 and so this is the final answer f prime of X is 20 X to the fourth power plus 12x cubed plus 9 now let&amp;#39;s say that f of X is 2x to the fifth power plus let&amp;#39;s say 5x to the third power plus 3x squared plus 4 and so you&amp;#39;re given this function and you&amp;#39;re told to find the slope of the tangent line at x equals 2 go ahead and try this problem anytime we need to find a slope with the tangent line first you need to find the derivative of the function that is f prime of X and then simply plug in the x value into that function so let&amp;#39;s determine F prime of X versus the derivative of X to the fifth power is 5x to the fourth power and the derivative of x cubed is 3x squared and the derivative of x squared is 2x and for the constant we just don&amp;#39;t need to worry about it so f prime of X is going to be 10 2 times 5 is 10 so 10 X to the fourth power 5 times 3 is 15 3 times 2 is 6 and so this is what we have now to calculate the slope and let&amp;#39;s replace X with Thun now Q to the fourth power if we multiply to 4 times 2 times 2 is 4 times 2 is 8 times 2 is 16 and then 2 squared that&amp;#39;s 4 and 6 times 2 is 12 so now we have F prime of two is equal to ten times 16 which is 160 15 times four that&amp;#39;s 60 and 160 plus 60 that&amp;#39;s 220 and so the final answer is going to be 232 so that&amp;#39;s the slope of the tangent line when X is equal to two now let&amp;#39;s say that f of X is 1 over X what is the derivative of that function so what is the derivative of 1 over X how would you go about fighting it for a situation like this you need to rewrite the function and so we need to do is take the X variable and move it to the top when you do that the exponent changes signed it&amp;#39;s gonna change from positive 1 it&amp;#39;s a negative 1 now at this point you could use the power rule so remember the derivative of X to the N is n X raised to the N minus 1 so in this case n is negative 1 and negative 1 minus 1 is negative 2 so we have negative 1 X to the negative 2 now once you have the derivative you need to rewrite it into a more proper form so let&amp;#39;s take the X variable and move it back to the bottom so our final answer will look like this it&amp;#39;s negative 1 divided by x squared and so that&amp;#39;s how you could find the derivative of a rational function now let&amp;#39;s say that f of X is 1 over x squared go ahead and find f prime of x for the sake of practice so take a minute and try that example just like before we&amp;#39;re going to rewrite the function so let&amp;#39;s move the X variable to the numerator of the fraction so f of X is equivalent to X raised to the negative 2 now at this point after you rewrite it now you can find the derivative using the so let&amp;#39;s take the constant I mean of the constant but the exponent move it to the front so our n value is negative two and then let&amp;#39;s subtract negative 2 by one negative two minus one is negative three and now let&amp;#39;s rewrite this expression by taking the X variable and moving it to the bottom if you&amp;#39;re wondering why I divided by one it&amp;#39;s the same thing negative two X to the negative 3 is the same as negative two X to the negative three over one I just like to write it in a fraction so you could see what I&amp;#39;m going to do next and that is moving the X variable to the bottom so now the exponent will change from negative three it&amp;#39;s a positive three and so this is the final answer so this is the derivative of 1 over x squared now let&amp;#39;s try another example so let&amp;#39;s say that f of X let&amp;#39;s try a harder example let&amp;#39;s say it&amp;#39;s 8 over X to the fourth power go ahead and work on that problem so let&amp;#39;s rewrite the function let&amp;#39;s move the X variable to the top and so this is gonna be 8 X raised to the negative fourth power and now let&amp;#39;s differentiate the function so we need to use the constant multiple rule in this case so it&amp;#39;s going to be eight times the derivative so I&amp;#39;m going to write that as times D over DX the derivative of x to the negative fourth power and so in this case using the power rule our and the value is negative 4 so it&amp;#39;s gonna be times negative 4x raised to the negative 4 minus 1 and so negative 4 minus 1 is negative 5 and 8 times negative 4 is negative 32 so the answer is negative 32 to the negative 5th power well let&amp;#39;s rewrite it though let&amp;#39;s not leave it like that so if we move the X to the bottom we could write the final answer fully simplified as a negative 32 divided by X to the fifth power now let&amp;#39;s talk about finding the derivative of radical functions for instance what is the derivative of the square root of x so what do you think we need to do now the first thing we need to do is rewrite this expression as a rational exponent so X is the same as X to the first power and if you don&amp;#39;t see an index number it&amp;#39;s always a 2 so this is equivalent to X raised to the 1/2 now in this form we can use the power rule so n is going to be 1/2 so it&amp;#39;s 1/2 X raised to the 1/2 minus 1 now we need to subtract a fraction by a whole number and so we need to get common denominators 1 is the same as 2 divided by 2 2 divided by 2 is 1 and 1 over to the -2 over 2 that&amp;#39;s gonna be negative 1 over 2 not because we have a negative exponent we need to rewrite this so right now the 1 is on top and the X is also on top the 2 is in the bottom of the fraction but now I need to take the X variable and move it to the bottom so this becomes 1 over 2 times X raised to the positive 1/2 and at this point I can rewrite the rational exponent as a radical so we know that X to the 1/2 is the square root of x so the final answer is 1 divided by 2 square root X and so this is it now let&amp;#39;s work on another example so let&amp;#39;s say that f of X is the cube root of x to the fifth power so what is f prime of X go ahead and try that so let&amp;#39;s begin by rewriting this expression so the cube root of x to the fifth power we can rewrite that as a rational exponent and it&amp;#39;s going to look like this it&amp;#39;s x raised to the 5 over 3 so this number here becomes the numerator of the rational exponent and the index number becomes the denominator of the fraction that we see here now let&amp;#39;s use the power rule so in this case n is a fraction it&amp;#39;s going to be 5 over 3 and then we&amp;#39;ll have X raised to the N minus 1 so that&amp;#39;s 5 over 3 minus 1 now just like before we need to get common denominators 1 is the same as 3 divided by 3 and so we have 5 over 3 the minus 3 over 3 5 minus 3 is 2 so that becomes 2 over 3 and I&amp;#39;m going to rewrite this as 5 times X raised to 2/3 divided by 3 now because the exponent is still positive we don&amp;#39;t need to move the X variable to the bottom and that&amp;#39;s not necessary the last thing that we need to do is convert this back into a radical expression so this is going to be 5 times the cube root of x squared over 3 and so this is the final answer here&amp;#39;s another one that you can work on so let&amp;#39;s say that we have the monomial or rather just a radical expression the 7th root of x to the fourth power what is the derivative of that expression so this is X raised to the 4 over 7 and let&amp;#39;s use the power rule so n is 4 over 7 and it&amp;#39;s gonna be X to the 4 over 7 minus 1 now to get common denominators let&amp;#39;s replace one with 7 over 7 now 4 over 7 minus 7 over 7 that&amp;#39;s a negative 3 over 7 so I&amp;#39;m gonna rewrite this as a fraction the fours on top the X variable is currently on top but the 7 is in the bottom of the fraction now in this case we do have a negative exponent so we need to move the X variable to the bottom and so it&amp;#39;s gonna be 4/7 times X raised to the 3 over 7 and now we can rewrite the rational exponent as a radical expression so the final answer is gonna be 4/7 times the seventh root of x cube and that&amp;#39;s it so this is the answer now of course if you want to you can rationalize the denominator but I&amp;#39;m not gonna worry about that in this video now let&amp;#39;s talk about some other problems that you might see in your homework so let&amp;#39;s say if you&amp;#39;re given a problem that looks like this it has x squared on the outside and then within a parenthesis it has X cubed plus 7 how would you find the derivative of this expression what would you do in this case the best thing to do right now with what you already know is to distribute the x squared 2x cubed plus 5 and then you can find the derivative so x squared times X cubed that&amp;#39;s going to be X to the fifth power because 2 plus 3 is 5 and then x squared times 7 is 7x squared so in this form it&amp;#39;s very easy to find the first derivative so the derivative of X to the fifth is 5x to the fourth and the derivative of x squared is 2x so the final answer is 5x to the fourth power plus 14x and so that&amp;#39;s what you need to do if you ever come across like a situation like that now let&amp;#39;s try a different example let&amp;#39;s say that f of X is equal to 2x minus 3 raised to the second power what would you do in this case now there&amp;#39;s something called the chain rule which we can use here but you haven&amp;#39;t learned that yet so we&amp;#39;ll save that for another day or rather later on in this video something we can do is expand this expression so whenever you see an exponent of 2 whatever that exponent is attached to it means that you have two of these things and multiplied to each other so this expression is equivalent to 2x minus 3 times another 2x minus 3 which that does not look like a 3 and so what we&amp;#39;re doing is we&amp;#39;re multiplying a binomial by another binomial and so let&amp;#39;s use the foil method so let&amp;#39;s multiply the first two terms 2x times 2x 2 times 2 is 4 x times X is x squared and then 2x times negative 3 that&amp;#39;s negative 6x negative 3 times 2x is also negative 6x and then we have negative 3 times negative 3 which is positive 9 now let&amp;#39;s combine like terms so negative 6x plus negative 6x is the negative 12x and now we could find the first derivative the derivative of x squared is 2x the derivative of x is 1 and for a constant is 0 so the final answer is going to be 8x minus 12 and so that&amp;#39;s the derivative of 2x and minus 3 we&amp;#39;re so that&amp;#39;s what you could do in a situation like this now let&amp;#39;s say we have a fraction X to the fifth plus 6x to the fourth power plus 5x cube divided by x squared in this case what is the derivative of f of X now based on the previous examples you know you need to simplify this before finding the derivative so how can you simplify this expression if you&amp;#39;re dividing a trinomial by a monomial what you could do is divide every term by x squared separately so let&amp;#39;s begin by dividing X to the fifth by x squared and it&amp;#39;s a point to understand that when you multiply let&amp;#39;s say x squared by X cube we need to add the exponents when you divide you need to subtract so this is 5 minus 2 that&amp;#39;s X cubed and so that&amp;#39;s going to be the first part so X to the fifth power divided by x squared is X cubed now 6 X to the fourth power divided by x squared that&amp;#39;s going to be 6 x squared all you need to do is subtract to the exponents 4 minus 2 is 2 and 5x cubed divided by x squared is going to be 5x to the first power because 3 minus 2 is 1 and so now we have this simplified polynomial and now we can find the first derivative so it&amp;#39;s going to be 3x squared and the derivative of x squared is 2x and the derivative of X is 1 so the final answer is 3x squared plus 12x plus 5 and that concludes this example now let&amp;#39;s talk about the derivatives of trigonometric functions and I&amp;#39;m going to give you a few that you need to know and for now write these downs because we&amp;#39;re going to use this later now the derivative of sine X you need to know is cosine X and the derivative of cosine X is negative sine X so that&amp;#39;s the first two you need to know next you need to know that the derivative of secant X is secant X tangent X and the derivative of cosecant X is actually very similar it&amp;#39;s going to be negative cosecant X cotangent X something that helps me to remember these things is that if you see a C in front you&amp;#39;re going to have a negative sign like the derivative of cosine it was negative sine now consider the last two the derivative of tangent is secant squared and based on that what do you think the derivative of cotangent X will be well notice that we do have the C it turns out that it&amp;#39;s negative cosecant squared X so keep those six derivative functions in mind because we&amp;#39;re gonna be using them later now the next thing that we&amp;#39;re going to go over is the product rule and here it is so let&amp;#39;s say if you have two functions multiplied to each other and you wish to find the derivative of that result it&amp;#39;s going to be the derivative of the first function times the second plus the first function times the derivative of the second so let&amp;#39;s say if we wish to determine the derivative of X times sine X so in this case we could say that f is x squared and G is sine X so I&amp;#39;m going to write it out so if f is x squared what is f prime F prime is the derivative of F and the derivative of x squared is 2x now G is going to be equivalent to sine X and G prime is the derivative of sine X which we now know is cosine X so now at this point all we need to do is basically plug in what we have on the right side of the equation so f prime is 2x G is sine X f is x squared and G prime is cosine X so this is the answer if you want to you can factor out the GCF which is X but I&amp;#39;m going to leave the answer like this and so that&amp;#39;s how you could use the product rule when finding the derivative of functions that are multiplied to each other now let&amp;#39;s try some other examples try this problem what is the derivative of let&amp;#39;s say 3x to the fourth power plus 7 times X cubed minus 5x now granted we can foil this expression because we did an example like that earlier but let&amp;#39;s use the product rule to get the answer feel free to pause the video if you want to so what I&amp;#39;m gonna do first is I&amp;#39;m gonna write the formula so the derivative of F times G is going to be the derivative of the first part times the second plus the first part let me write that again times the derivative of the second so what&amp;#39;s F and what&amp;#39;s G we&amp;#39;re going to say that F is the first part and so we&amp;#39;re going to say that f is 3x to the fourth power plus seven so what is f prime so the derivative of X to the fourth is 4x cubed but we&amp;#39;re going to multiply that by three and so three times four that&amp;#39;s going to give us 12 so this is going to be 12x cubed and the derivative of seven is zero now G is the second part of the function so G is X cubed minus 5x G prime is going to be three x squared and the derivative of X is one times negative five so that&amp;#39;s just going to be negative five and so that&amp;#39;s G prime so using the formula this expression becomes equal to which I&amp;#39;m going to write over here this is going to be F prime which is 12x cubed times G which is X cubed minus 5x plus F and that&amp;#39;s three X to the fourth plus seven times G prime 3x squared minus five and that&amp;#39;s it right there now here is a challenge problem for you what is the derivative of x cubed times tangent x times three x squared minus nine so this time we have three parts being multiplied to each other so we saw how to use the product rule when having two different functions being multiplied to each other but what about three different functions so if the derivative of let&amp;#39;s say a two-part function like F times G if that&amp;#39;s F Prime and G Plus F G Prime what would the derivative of let&amp;#39;s say F times G times H be so this is going to be we&amp;#39;re going to differentiate the first part and then leave the second two parts the same plus we&amp;#39;re going to leave the first part this differentiate the second part and then leave the third part the same and then it&amp;#39;s going to be the first two parts times the derivative of the last part so when using the product rule when you differentiate one part the other two parts should remain the same and then you just go in order differentiate the first part and then the second part and then the third part so once you understand the format or the procedure of doing this you can just go ahead and get the answer without actually writing down what&amp;#39;s F of G and H so first let&amp;#39;s find the derivative of the first part the derivative of x cube is 3x squared now the other two parts the G and H we&amp;#39;re just going to rewrite it for now so it&amp;#39;s going to be times tangent X and then times 3x squared minus 9 now let&amp;#39;s rewrite the first part which is X cube and then we&amp;#39;re going to take the derivative of the second part the derivative of tangent if you remember is secant squared now let&amp;#39;s rewrite the third part which is 3x squared minus 9 now for the last part we&amp;#39;re going to rewrite the first two parts X cube and tangent X but this time we are going to take the derivative of 3x squared minus 9 which is going to be just 6x because this will go to 0 now let&amp;#39;s say if we want to find the derivative of a fraction such as let&amp;#39;s say 5x plus 6 divided by 3x minus 7 in this case you want to use something called the quotient rule and here&amp;#39;s the formula that we&amp;#39;re going to use so the derivative of let&amp;#39;s say F divided by G this is going to be G F - f G prime divided by G squared and this is something that you simply need to commit to memory you just got to know that function at least it works for me when I was in high school I just memorized that function now F is going to be the top portion of this function so in this case F is going to be 5x + 6 F prime is the derivative of F so the derivative of 5x + 6 is 5 now G G is going to be the bottom part of this function so G is going to be 3x - 7 which means G Prime history so if it helps to write everything out by all means go ahead and do that if it makes your life easier or if it helps you to avoid mistakes and on a test one of the biggest things that you have to do is avoid mistakes because if you make a mistake even if you know it I mean that&amp;#39;s just gonna ruin your test score now let&amp;#39;s go ahead and finish this G is 3x minus 7 and then F prime that&amp;#39;s 5 and then we have F which is 5x plus 6 and then G prime that&amp;#39;s 3 and then divided by G squared so G is 3x minus 7 and then let&amp;#39;s square that now in this case I&amp;#39;m going to simplify because it doesn&amp;#39;t require that much work to do so so let&amp;#39;s begin by distributing the 5 to 3x - 7 so 5 times 3x that&amp;#39;s 15 X and then 5 times 7 or 5 times negative 7 rather that&amp;#39;s negative 35 and here we have negative 5x times 3 which is going to be negative 15 x + 6 times 3 is 18 but we got the negative sign so that&amp;#39;s going to be negative 18 and I&amp;#39;m in the foyer the stuff on the bottom because it looks better this way now we could cancel 15 X negative 35 minus 18 that&amp;#39;s going to be negative 53 and so the answer is negative 53 divided by 3x minus 7 squared if it&amp;#39;s easy to simplify your answer feel free but sometimes if it takes a lot of work to simplify it most teachers will allow you just to write the answer the way it is some teachers will allow you to leave the answer like this so you need to basically know your teacher and how they want you to write the final answer

Transcript for:Derivatives Lecture Notes

Transcript for:
Derivatives Lecture Notes