in this lesson we're going to focus on finding the derivative of a function so let's start with a constant the derivative of any constant is equal to zero so for instance the derivative of the constant 5 is zero and the derivative let's say of -7 is also zero 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's say if we have the function f ofx is equal to 8 if we were to graph this function it would look like a straight line at yal 8 so around this region let's say that's uh yal 8 now what is the slope of a straight line the slope of a straight line is zero and so if you were to find the derivative of this function represented by fim of X that will give you zero if you see D over DX it means that you're about to differentiate something with respect to X so this means the derivative of f ofx is equal to frime of X now how can we find the derivative of a monomial for example what is the derivative of x^2 now there's something called the power rule and the power rule is very useful for finding the derivative of monomials so here'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 * X ra to the nus1 and so that's the formula 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^2 is going to be 2 X to 2 - 1 which is one or basically 2 to X power so that's the derivative of x^2 now let'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 cub x to the 4th and also X to the 5ifth power so the derivative of x Cub is going to be 3 in this case n is 3 so it's 3x raised to 3 - 1 and so 3 - 1 is 2 so the answer is going to be 3 x^2 now the derivative of x X the 4th power in this case n is 4 so it's going to be 4 x raed to 4 - 1 and 4 - 1 is 3 so it's 4X Cub now for the last one in this case n is 5 so it's going to be 5 x raised to the 5 - 1 and 5 - 1 is 4 so it's 5 x 4th power and so that's a simple way in which you could find the derivative of a function now let's say if you want to find the derivative of 4 x to the 7th 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 let's say a it's going to be the constant times the derivative of that monomial or that function let's just put F ofx here so in this case our C value is four and F ofx is X to 7th so I'm just going to color C coded so C is 4 as you could see here and F ofx is X 7th so let's bring out the four to the front and then we're going to multiply it by the derivative of x to the 7th Now using the power rule we can find the derivative of that function so it's going to be 7 x raised to the 7 - 1 and so 7 - 1 is 6 so we have 7 x to the 6 power and now we can multiply 4 and 7 4 * 7 is 28 so the answer is going to be 28 x to the 6 power and so this is it now let's try some more examples so go ahead and find the derivatives of these two monomials 8X to the 4th and also the derivative of 5 x ra 6 power so go ahead and take a minute so what we're going to do is move the constant to the front and using the power rule we're going to differentiate X to 4th so the derivative of x to 4th is going to be 4 * X raed to 4 - 1 and 4 - 1 is 3 and now we need to multiply 8 by 4 8 * 4 is 32 so the answer is 32 x to the 3 power now for the next one let's move the constant to the front and then we're going to multiply by the derivative of x to the 6 power so we can take this exponent move it to the front so this is going to be 5times the derivative of x 6 which is 6 x to 6 - 1 which is 5 so 6 x 5th power and now let's multiply 5 * 6 5 * 6 is 30 so it's 30 x to the 5th power now for the sake of practice let's try a few more examples so let's try the derivative of 9x to fifth power and also the derivative of 6 x to the 7th power so go ahead and try those two examples so this is going to be 9 time the derivative of x the 5th power and the derivative of x to the 5th power is 5 x to the 4th power and so 9 * 5 that's going to be 45 so the answer is 45 X to 4th power and so that's it for this one now for the last one it's going to be 6 * the derivative of x to the 7th power and the derivative of x to the 7th power is going to be we could take the seven move it to the front so that's going to be 7 x 7 - 1 which is 6 and 6 * 7 is 42 so it's 42 x raised to the 6 power now we said that the derivative of x^2 is equal to 2x now how do we know that by the way let's say if f ofx is X2 that means that the derivative of f ofx which is frime 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'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's referred to as the definition of the derivative perhaps you've seen this function frime of X is equal to the Limit as H approaches zero of f ofx plus hus FX / H now you might be wondering what exactly is f ofx plus h what is that well we know what f ofx is f ofx is x^2 so if f ofx is x^2 what is f ofx + H all you need to do is replace the X with X+ H so inside here we had x^2 so instead of X we're going to replace it with x + H so f of x + H is x + H2 now let's plug it into this formula so let me erase this so remember we're trying to show that frime of X is equal to 2x I'm going to erase that soon right now we have this frime of X is equal to the Limit as H approaches zero and this right here we know it's x + h^ 2 minus F ofx which is x^2 all divid H now what do you think we need to do at this point what's our next step our next step is to foil that expression so this is equivalent to the Limit as H approaches Z and x + h^ 2 is the same as x + H * x + H so let's go ahead and foil that expression now when taking 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't rewrite it so here we have x * X and that's going to be x^2 and then we have x * H and then it's h * X which is the same as x * H and then the last one H * H so that's h^ 2 and then minus x^2 / H now at this point we can cancel the x s term and we can combine like terms xh plus xh that's 2 xh so now we have 2 xh plus h^2 / H now our next step is to factor the GCF that is is the greatest common factor which is H so if we take out an H from 2x H that's going to be 2X and h^2 / 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 zero of 2x + H and so when H becomes zero this is going to be 2x + 0 so basically H disappears as H approaches Z and that's how we get the final answer 2X and so that's why the derivative of x^2 is 2X and so that's how you can find this answer using the limit process now we said that the derivative is a function that will give you the slope at any x value so let's say that f ofx is x^2 and we wish to find the slope of the tangent line at x = 1 so we know what frime of X is using the power rule it's 2X and so to find a slope at x = 1 we need to evaluate frime of X when when X is 1 and so that's going to be 2 * 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 = x^2 it will look something like this and when X is equal to 1 the slope of the tangent line will equal 2 so I'm going to put Mt or M tan so the slope of the tangent line is 2 when X is equal to 1 and so that's what the derivative function tells you it gives you the slope of the tangent line at some x 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's uh do that again and a 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've learn that to find the slope of a line you need two points and this is basically finding the slope of a secant line that's on a curve so let's put M secant and you you known it as Y2 - y1 = X2 - X1 so we could take two points on this curve and basically get a secant line and as those two points approach 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 = 1 so we can choose let's say .9 and 1.1 as our X1 and X2 values because if you add up those two numbers and divide by two the average of 0.9 and 1.1 is 1 or we could pick 0.99 and 1.01 because the midpoint of those two numbers is still one however 099 and 1.01 is closer to 1 than 0.9 and 1.1 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 xal 1 and so let's go ahead and calculate those values so let's say that X1 is .9 to begin with and X2 is 1.1 and let's use this formula to calculate the slope of the secant line now keep in mind the slope of the tangent l line is this number it's equal to two so this is going to be Y 2 - y1 / X2 - X1 and so Y2 corresponds to the Y value for this x value and Y is equal to F ofx so we could use this function FX = x^2 to find Y2 so when X2 is 1. 1 Y 2 is 1.12 because y = x^2 when X1 is .9 y1 is9 2 now 1.1 squar that's 1.21 and .9 2ar is 81 and 1.1 - .9 is2 1 . 21 -81 / .2 gives us already an exact answer which is two and so there's no need to use 099 in this instance we can see that it's exactly the same so let's try an example where it may not be exactly the same so this time let's say that F ofx is X cub and we wish to calculate the slope of this hand line at x = 2 so we know what frime of X is the derivative of x Cub using the power rule is 3x2 so the derivative at x = 2 is going to be 3 * 2^ 2 2^ 2 is 2 * 2 that's 4 * 3 is 12 so the slope of the tangent line at xal 2 is 12 now let's see if we can approximate this value with the slope of the secant line so let's choose an X1 value of 1.9 and an X2 value of 2.1 and so the slope of the secant line between those two points is going to be Y 2 minus y y1 over X2 - X1 so in this case we said X2 is 2.1 now what's Y2 Y2 has to be 2.1 ra the 3 power because Y is equal to X Cub X1 is 1.9 so y1 is 1.9 to the 3 power now 2.1 raed to the 3r power that's going to be 9261 and 1.9 raised to the 3 power that's going to be 6. 859 and 2.1 minus 1.9 that's .2 9.26 1 minus 6.85 that's 2.42 and if we divide that by 0.2 it gives us a very good approximation actually 12.01 and so you could see that you can approximate the slope with this tent line using the slope of the secant line and that's what the derivative tells you it gives you the slope of the tangent line which touches the curve at one point that's some x value in review remember this the derivative is a function that helps you to find the slope of a tangent line at some value of x so keep that in mind now let's talk about finding the derivative of a polom function so let's say that F ofx is X Cub + 7 x^2 - 8x + 6 what is the derivative of that function so what is frime 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 XB is 3x2 now what about the derivative of 7 x^2 using a constant multiple rule it's going to be 7 * the derivative of x^2 which is 2X or 2x the first Power and 7 * 2x is 14x now what about the derivative of -8x what is that equal to so keep in mind this is -8 * x to the 1 power so this is going to be -8 * the derivative of x to the first Power and what is the derivative of x to the first Power well using the power rule we need to move the one to the front so 1 * X ra 1 - 1 which becomes -8 * 1 x 0 now what is X 0 x to the 0 or anything raised to the 0 power is 1 so this becomes 8 * 1 which is just8 so the derivative of 8X is simply 8 and the derivative of any constant is zero so we could stop it here this is the answer frime of X is 3x^2 + 14x - 8 the video that you'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'm going to 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's get back to the lesson now let's say that f ofx is 4 x 5th power + 3 x 4th power + 9x - 7 what is frime of X so go ahead and try this for the sake of practice so using the constant multiple rule we're going to rewrite the constant and take the derivative of x to the 5th power using the power rule and so that's going to be 5 x 4th power now the derivative of x 4th is 4X CU and the derivative of x is always just one and the derivative of a constant is zero and so we have 4 * 5 which is 20 and 3 * 4 which is 12 and so this is the final answer frime of X is 20 x to 4th power + 12 x Cub + 9 now let's say that f ofx is 2 X to the 5th power plus let's say 5 x to the 3 power + 3x^2 + 4 and so you're given this function and you're told to find the slope of the tangent line at x = 2 go ahead and try this problem anytime you need to find the slope of the tangent line first you need to find the derivative of the function that is frime of X and then simply plug in the x value into that function so let's determine frime of X first the derivative of x 5th power is 5 x 4th power and the derivative of x Cub is 3x² and the derivative of x^2 is 2X and for the constant we just don't need to worry about it so frime of X is going to be 10 2 * 5 is 10 so 10 x 4th power 5 * 3 is 15 3 * 2 is 6 and so this is what we have now to calculate the slope let's replace x with 2 now 2 to the 4th power if we multiply 2 four times 2 * 2 is 4 * 2 is 8 * 2 is 16 and then 2^ 2 that's 4 and 6 * 2 is 12 so now we have frime of 2 is equal to 10 * 16 which is 160 15 * 4 that's 60 and 160 + 60 that's 220 and so the final answer is going to be 232 so that's the slope of the tangent line when X is equal to two now let's say that f ofx is 1 /x what is the derivative of that function so what is the derivative of 1 /x how would you go about finding it for a situation like this you need to rewrite the function and so what you need to do is take the X variable and move it to the top when you do that the exponent changes sign it's going to change from positive 1 to 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 nus1 so in this case n is -1 and -1 - 1 is -2 so we have -1 x -2 now once you have the derivative you need to rewrite it into a more proper form so let's take the X variable and move it back to the bottom so our final answer will look like this it's -1 / x^2 and so that's how you could find the derivative of a rational function now let's say that f ofx is 1 x^2 go ahead and find frime of X for the sake of practice so take a minute and try that example just like before we're going to rewrite the function so let's move the X variable to the numerator of the fraction so f ofx is equivalent to X raised to the -2 now at this point after you rewrite it now you can find the derivative using the power rule so let's take the constant I mean not the constant but the exponent move it to the front so our n value is -2 and then let's subtract -2 by 1 -2 - 1 is -3 and now let's rewrite this expression by taking the X a variable and moving it to the bottom if you're wondering why I divided by one it's the same thing -2X -3 is the same as -2X -3 over 1 I just like to write it in a fraction so you could see what I'm going to do next and that is moving the X variable to the bottom so now the exponent will change from -3 to pos3 and so this is the final answer so this is the derivative of 1x^2 now let's try another example so let's say that F ofx let's try a harder example let's say it's 8 over X to 4th power go ahead and work on that problem so let's rewrite the function let's move the X variable to the top and so this is going to be 8 x raised to the4 power and now let's differentiate the function so we need to use the constant multiple rule in this case so it's going to be 8 times the derivative so I'm going to write that as times d/ DX the derivative of X to the4 power and so in this case using the power rule our n value is4 so it's going to be time -4 x raised to4 -1 and so -4 - 1 is-5 and 8 * 4 is -32 so the answer is -32 X to theth power but let's rewrite it though let's not leave it like that so if we move the X to the bottom we can write the final answer fully simplified as -32 / X 5th power now let'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't see an index number it's always a two so this is equivalent to X rais to the2 now in this form we can use the power rule so n is going to be/ 12 so it's 12 x raised to the 12 - 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 / 2 2 / 2 is 1 and 1 / 2 - 2 over 2 that's going to be -1 /2 now because we have a negative exponent we need to rewrite this so right now the one is on top and the x is also on top the two 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/ 2 * X raised to the positive 12 and at this point I can rewrite the rational exponent as a radical so we know that x^2 is theare root of x so the final answer is 1 divid 2 < TK X and so this is it now let's work on another example so let's say that f ofx is the cube root of x to the 5th power so what is frime of X go ahead and try that so let'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's going to look like this it's X raed to 5 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's use the power rule so in this case N is a fraction it's going to be 5 over 3 and then we'll have X raised to the nus1 so that's 5 3 - 1 now just like before we need to get common denominators 1 is the same as 3 / 3 and so we have 5 over 3 - 3 over 3 5 - 3 is 2 so that becomes 2 over 3 now I'm going to rewrite this as 5 * X raised to 2/3 / 3 now because the exponent is still positive we don't need to move the X variable to the bottom and that'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 * the cube root of x^2 over 3 and so this is the final answer here's another one that you can work on so so let's say that we have the monomial or rather just the radical expression the 7th root of x 4 power what is the derivative of that expression so this is X raed to the 4 7 and let's use the power rule so n is 4 7 and it's going to be x to the 4 7 -1 now to get common denominators let's replace 1 with 7 over 7 now 4 7 - 7 over 7 that's -3 over 7 so I'm going to rewrite this as a fraction the four is on top the X variable is currently on top but the seven 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's going to be 4 / 7 * X raed to 3 7 and now we can rewrite the rational exponent as a radical expression so the final answer is going to be 4 / 7times the 7th root of x cub and that's it so this is the answer now of course if you want to you can rationalize the denominator but I'm not going to worry about that in this video now let's talk about some other problems that you might see in your homework so let's say if you're given a problem that looks like this it has X2 on the outside and then within a parenthesis it has X Cub + 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^2 to X Cub + 5 and then you can find the derivative so x^2 * X Cub that's going to be X to 5th power because 2 + 3 is 5 and then x^2 * 7 is 7 x^2 so in this form it's very easy to find the first derivative so the derivative of x X 5th is 5 x 4th and the derivative of x^2 is 2x so the final answer is 5X 4th power + 14x and so that's what you need to do if you ever come across like a situation like that now let's try a different example let's say that f of x is equal to 2x - 3 raised to the 2 power what would you do in this case now there's something called The Chain rule which we can use here but you haven't learned that yet so we'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 two whatever that exponent is attached to that means that you have two of these things multiply to each other so this expression is equivalent to 2x - 3 * another 2x - 3 which that does not look like a three and so what we're doing is we're multiplying a binomial by another binomial and so let's use the foral method so let's multiply the first two terms 2x * 2x 2 * 2 is 4 x * X is x^2 and then 2x * 3 that's -6x -3 * 2x is also -6x and then we have -3 * -3 which is POS 9 now let's combine like terms so -6x + -6x is -12x and now we could find the first derivative the derivative of x^2 is 2x the derivative of x is 1 and for a constant is zero so the final answer is going to be 8x - 12 and so that's the derivative of 2x - 3^ 2 so that's what you could do in a situation like this now let's say we have a fraction X 5th + 6 x 4 power + 5xb / X s in this case what is the derivative of f ofx now based on the previous examples you know you need to simplify this before finding the derivative so how can we simplify this expression if you're dividing a trinomial by a monomial what you could do is divide every term by X2 separately so let's begin by dividing x to the 5th by x^2 and it's important to understand that when you multiply let's say X by X Cub we need to add the exponents when you divide you need to subtract so this is 5 - 2 that's X cub and so that's going to be the first part so x to the 5th power / x^2 is X Cub now 6 x 4 power / x^2 that's going to be 6 x^2 all you need to do is subtract the exponents four - 2 is 2 and 5 x Cub / x^2 is going to be 5 x to the 1 power because 3 - 2 is 1 and so now we have this simplified polinomial and now we could find the first derivative so it's going to be 3x^2 and the derivative of x^2 is 2X and the derivative of x is 1 so the final answer is 3x^2 + 12x + 5 and that concludes this example now let's talk about the derivatives of trigonometric functions and I'm going to give you a few that you need to know and for now write these down uh because we're going to use this later now the derivative of sinx you need to know is cine X and the derivative of cosin x is sinx so that'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 cose X is actually very similar it's going to be negative cose xent X something that helps me to remember these things is that if you see a c in front you're going to have a negative sign like the derivative of cosine it was negative sign now consider the last two the derivative of tangent is secant squar and based on that what do you think the derivative of coent x will be well notice that we do have the C it turns out that it's negative cose 2 x so keep those six uh derivative functions in mind because we're going to be using them later now the next thing that we're going to go over is the product rule and here it is so let's say if you have two functions multiply to each other and you wish to find the derivative of that result it's going to be the derivative of the first function times the second plus the first function time the derivative of the second so let's say if we wish to determine the derivative of x^2 * sin x so in this case we could say that f is x^2 and G is sinx so I'm going to write it out so if f is x^2 what is f Prime F Prime is the derivative of F and the derivative of x^2 is 2x now G is going to be equivalent to sinx and G Prime is the derivative of sinx 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 fime is 2x G is sin x f is x^2 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'm going to leave the answer like this and so that's how you could use the product rule when finding the derivative of functions are multiplied to each other now let's try some other examples try this problem what is the derivative of let's say 3x to the 4th power + 7 * X Cub - 5x now granted we can foil this expression because we did an example like that earlier but let's use the product rule to get the answer feel free to pause the video if you want to so what I'm going to do first is I'm going to write the formula so the derivative of f * 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's F and what's G we're going to say that f is the first part and so we're going to say that f is 3x 4 power + 7 so what is frime so the derivative of x to the 4th is 4X CU but we're going to multiply that by three and so 3 * 4 that's going to give us 12 so this is going to be 12 x cub and the derivative of 7 is zero now G is the second part of the function so G is X Cub - 5x G Prime is going to be 3x^2 and the derivative of x is 1 * 5 so that's just going to be5 and so that's G Prime so using the formula this expression becomes equal to which I'm going to write over here it's going to be F Prime which is 12x cub time G which is X Cub - 5x + f and that's 3x 4 + 7 * G Prime 3x^2 - 5 and that's it right there now here is a challenge problem for you what is the derivative of x Cub * tangent x times 3x^2 - 9 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's say a two-part function like f * G if that's F Prime G plus FG Prime what would the derivative of let's say f * G * H be so this is going to be we're going to differentiate the first part and then leave the second two parts the same plus we're going to leave the first part the same differentiate the second part and then leave the third part the same and then it'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's f g and H so first let's find the derivative of the first part the derivative of x Cub is 3x^2 now the other two parts G and H we're just going to rewrite it for now so it's going to betimes tangent X and then time 3x^2 - 9 now let's rewrite the first part which is X cub and then we're going to take the derivative of the second part the derivative of tangent if you remember is secant squ now let's rewrite the third part which is 3x^2 - 9 now for the last part we're going to rewrite the first two parts X cub and tangent X but this time we are going to take the derivative of 3x^2 - 9 which is going to be just 6X because this will go to zero now let's say if we want to find the derivative of a fraction such as let's say 5x + 6 ID 3x - 7 in this case you want to use something called the quotient Rule and here's the formula that we're going to use so the derivative of let's say F / G this is going to be g f F Prime minus f g Prime / G ^2 and this is something that you simply need to commit to memory you just got to know that function at least it worked 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 is 3 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's just going to ruin your test score now let's go ahead and finish this G is 3x - 7 and then F Prime that's five and then we have F which is 5x + 6 and then G Prime that's three and then divided by G ^2 so G is 3x - 7 and then let's square that now in this case I'm going to simplify because it doesn't require that much work to do so so let's begin by Distributing the 5 to 3x - 7 so 5 * 3x that's 15x and then 5 * 7 or 5 * 7 rather that's -35 and here we have NE 5x * 3 which is going to be -5x and 6 * 3 is 18 but we got the negative sign so that's going to be 8 and I'm not going to foil the stuff on the bottom because it looks better this way now we could cancel 15x -35 - 18 that's going to be 53 and so the answer is 53 / 3x - 7^ 2 if it'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