in this video we're going to focus on finding the average rate of change and the instantaneous rate of change so let's say if we have the function f ofx is equal to X raised to the 3 power how can we calculate the average rate of change on the interval from 1 to three now what we need to realize is that the average rate of change represents the slope of the secet line so here's the graph x Cub or Y is equal to X Cube Y is the same as F ofx so at the point one and three which is these two points the slope of the secant line is the average rate of change the instantaneous rate of change is the SL of the Tang line so let's say if we wanted to find the instantaneous rate of change at xals 2 it would represent the slope of the tangent line the tangent line is a line that touches the curve at one point a secant line touches the curve at two points notice that you can approximate the slope of the the tangent line by finding the slope of the secant line if you want to find the slope of the tangent line at two you can estimate it using the slope of the seant line between one and three because two is right in the middle between one and three so for this function go ahead and calculate the average rate of change the average rate of change which is the slope of the seeet line is the change in y divid by the change in X now in algebra you've learned that the slope is basically Y 2 - y1 / X2 - X1 so the average rate of change let's just put average this is going to be F of B minus F of a keep in mind f of x is the same thing as y / B minus a which is the change in the X vales so in calculus this is the formula that you want to remember that's the average rate of change so for this particular problem a is 1 and B is three so we're looking for f of three minus F of 1 / 3 - 1 so if we plug in three into the equation 3 the 3r power is 27 1 the 3r that's 1 * 1 * 1 that's simply 1 1 3 - 1 is 2 so it's 26 over 2 and so it's 13 that is the average rate of change in the interval up between 1 to three so that is the slope of the secant line now what is the instantaneous rate of change at xal 2 the instantaneous rate of change is represented by frime of 2 that's what it means it's basically the derivative of the function at xal 2 which represents the slope of the tangent line but how can we estimate this value using the average rate of change equation so if you want to estimate frime of 2 using the average rate of change you want the interval to be smaller you want it to be closer to two so let's see what happens if we decrease the range from 1.9 to 2.1 notice that two is in the middle between 1.9 and 2.1 but 1.9 and 2.1 is closer to two than 1 and three so fime of 2 is approximately going to be F of 2.1 minus F of 1.9 / 2.1 minus 1.9 so so what is f of 2.1 what's 2.1 raised to the 3 power okay for this we need a calculator it's going to be 9261 and then 1.9 raised to the 3 power is about 6859 2.1 minus 1.9 that's. 2 so 9.26 - 6.85 9 / .2 so frime of 2 is approximately 12.01 which is pretty close to 12 so notice that once we um calculated the average rate of change from 1 to three it was about 13 but from 1.9 to 2.1 it's 12.01 12 is not too far from 13 so the instantaneous rate of change is very close to 12 but now let's see what happens if we decrease the range so let's make it even closer to two so let's calculate the average rate of change from 1.99 to 2.01 so this should be very very close to the instantaneous rate of change at two so let's find out what F of 2.01 is minus F of 1.99 / 2.01 - 1.99 2.01 raised to the 3 power so I'm going to write the exact number that is equal to 8.12 61 and 1.99 raised to the third power that's 7.88 0599 2.01 minus 1.99 that's 02 now if you type the whole thing in 8.1 12061 minus 7.88 0599 you should get 24 and if you divide it by 02 your answer is about 12.01 so as you can see as you get closer to two the answer converg to 12 so we can say with good confidence that frime of 2 is 12 that is the slope of the tangent line when X is equal to 2 so as you can see when the interval When A and B becomes very close to 12 the average rate of change approaches the instantaneous rate of change so we can come up with an expression frime of X is equal to the Limit as X approaches zero for the change in y divid by the change in X so this is the slope that help us to calculate the average rate of change but the average rate of change becomes the instantaneous rate of change or the derivative of the function when Delta X approaches zero in the last example when a was 1.99 and B was 2.01 Delta X was very small it was 02 which is pretty close to zero and notice that when X was when Delta X was very small the answer converged to 12 so this is how you can find the derivative of a function at point now perhaps you've seen this formula limit as H approaches zero f ofx + hus F ofx / H so if you think about it this equation means the exact same thing as this one Delta Y is the change in y and we know that y represents F ofx so you could see the change in F ofx this is like f of B minus F of A and B minus a is like H which is Delta X so basically it's the slope equation but as the denominator approaches zero here we have Delta X approaching zero and H approaches zero which is the same as Delta X so you'll see this equation in your calculus course a lot and then there's the alternate form of the derivative the limit as X approaches a f ofx minus F of a over x - A this will give you fime of a so that's the slope of the tangent line frime of X is the derivative as a function and that function gives you the slope at any point in the curve fime of a gives you the slope of the tangent line at some point x but this gives you a function in terms of X where if you plug in X you could find a slope of this Tang line at any point now you need to know the easy way of finding the derivative and also how to use it with limits so let's go over the power rule what is the derivative of x to 4th power the derivative of x 4th power is 4X Cub now what is the derivative of x to the 5th power the derivative which is represented by FR Prime is 5 x 4th notice the pattern now what is the derivative of x to 6 this is going to be 6 F 5th now what about the derivative of x 7th what is the answer this is 7 x to the 6 power and the derivative of x to 8 is 8 x 7th now what about the derivative of 5 x 4th so what if you have a constant in front ignore the constant for now and differentiate x to the 4th power the process of finding the derivative is known as differentiation if you differentiate X the 4 it's 4X cub and then you multiply the 5 and the four this will give you 20 x Cub so what is the derivative of 6 x to the 7th power 4X cub and 5 x to the 8th power feel free to pause the video as you work on these examples so for the next one the derivative of x 7 is 7 x 6 and then 6 * 7 is 42 the derivative of x Cub is 3x^2 and then 4 * 3 is 12 the derivative of x 8 is 8 x^ 7 and 5 * 8 is 40 so it's 40 x 7 now what about the D derivative of x 5x -4x and 7 x the derivative of x to the first power is 1 X to 0 anything raised to the 0 power is one so this whole thing is one so for this one it's 5 * X the 1 power which becomes 5 * 1 x z which is simply five so for -4x it's simply -4 and for 7x the derivative is simply s now what about the derivative of a constant like 8 the derivative of a constant is always zero so the derivative of 5 is zero the derivative of Pi is zero the derivative of e without the x is zero now let's understand why we know the derivative of s is zero 7 is the same as 7 x to 0 x to0 is 1 so if you use the power rule if you move the zero to the front this is going to be 7 0 x to 1 anything times Z is zero so you'll end up getting zero now let's make sense of it let's say if we have the function y is equal to 3 so at three we have a horizontal line now remember the derivative represents the slope so if f ofx is equal to 3 D derivative is zero D derivative of a constant is zero if you think about it the slope of a horizontal line is equal to zero so that makes sense now let's say if we have the function y is = 2x - 5 to graph such a function the Y intercept is5 this is in y = mx + b form the slope is the number in front of X which is two so the rise over run is two over one if you go up two units from the first point you need to go one to the right so we have the13 up two over one we have the 21 and then up two over one 3 1 and then we can connect these points with a line so notice the slope of the line is always two so if f ofx is 2x - 5 what's frime of X the derivative of 2x is 2 in the last example we said the derivative of 8 x is 8 5x is 5 pix is pi 4X is 4 so the derivative of 2x is 2 the derivative of a constant is zero so notice that the derivative is always two at any point along this line the slope is always two a straight line has a constant slope now let's go back to our old example where we said f ofx is equal to x 3 power so remember remember we found the instantaneous rate of change which was about 12 and that represent the slope of the tangent line at xal 2 so here's another way you can find it so if we calculate frime of X we need to move the three to the front so it's going to be 3 x and then subtract this number by one so 3x squ this is the derivative as a function the derivative is a function that tells you the slope of the tangent line at any point x and the slope of the tangent line is the instantaneous rate of change so anytime you want to find the instantaneous rate of change you can simply find the derivative of the function and plug in the x value now if we want to find it at x = 2 simply replace x with 2 2^ 2 is 2 * 2 which is 4 3 * 4 is 12 so notice this is the exact answer that we got um when using the average rate of change to estimate the instantaneous rate of change when X became small or Delta X when the change in X became small the average rate of change got very close to 12 now let's use limits to get the same answer so if you want to find fime of X as a function use this equation the limit as H approaches z f ofx + H - F ofx / H now remember we know the answer frime of X as a function is 3x^2 when F ofx is X Cub so let's get the same answer using limits so F ofx is X Cub which is this portion of the function so we can put that here this is going to Bex Cub now F ofx + H instead of writing X we need to write x + H and then divid by H so now how can we simplify this expression there's two things that we can do we can either multiply x + H three times or we can factor using the difference of cubes and I prefer to use that equation the difference of Cubes is a the 3 minus B the 3 is a minus B time a 2 + a + b^ 2 so x + H is Like A and B is X so what we have is the limit as H approaches zero a minus B which is x + H that's a minus B which is - x a 2 is x + h 2 plus AB which is x + H * X plus b^ 2 which is x s divid h so notice that we can can cancel the X variables so now we have this expression H * x + h^ 2 + x + H * x + x^2 / H as soon as we get rid of the H on the bottom we can now plug in the limit into the equation so as H approaches zero everywhere we see an H we can replace it with zero at this point so once you plug in zero for H you don't need to write the limit expression anymore so it's going to be x + 0^ 2 + x + 0 * x + x^2 x + 0 is X so what we have there is x^2 and then x + 0 is also X so that's x * X Plus another X2 so it's X2 + X2 + X2 which is 3x2 which we already had that answer so we could see frime of X is 3x^2 by using the limit process so that's how you can get the derivative as a function using limits now how can we find frime of two using limits so keep in mind we know that frime of X is 3x^2 and frime of 2 if you plug into two it's equal to 12 but how can we get this answer directly using limits to do that there's an equation that looks like this frime of a is equal to the Limit as X approaches a f ofx - F of a / x - A so this equation comes from the average rate of change equation which is basically the slope of the secet line F of B minus F of A over B minus a the only difference is instead of X you have B and notice as X approaches a if if x is very close to a then xus a is almost zero and whenever the denominator is close to zero the average rate of change becomes the instantaneous rate of change so that's where this equation comes from it's very similar to finding the slope of the cat line so let's use it to find fime of two so in this problem a is equal to 2 now we know F ofx is X Cub what is f of a so if F ofx is X Cub what is the value of f of two F of two is going to be 2 to the 3 power which is equal to 8 so let's put an8 on the bottom we have x - A which is x - 2 so we need to factor x Cub minus 8 using the difference of Cubes equation as we used it before so in this problem X is going to represent A and B is going to be the cube root of 8 which is two so we have the limit as X approaches 2 a - b which is x - 2 * a^ 2 which is x 2 a that's x * 2 or 2X and b^ 2 is 2^ 2 which is 4 divided by xus 2 so at this point we can cancel x - 2 once we do that we can replace two or plug in two into the equation at that point we no longer need to write the limit expression so let's plug in two so it's going to be 2^2 + 2 * 2 + 4 which is 4 + 4 + 4 4 + 4 + 4 three times is the same as multiplying 4 by 3 since you're add in four three times and this equals 12 which is the slope of the tangent line at xal 2 so that's how you can find the value of the derivative at a point if you want to find the slope with the tangent line you can use the this form of the uh derivative equation now let's put everything together all within one practice problem so let's say f ofx is = to x^2 + 2x - 3 so go ahead and find the average rate of change between 2 and four so the average rate of change is f of B - F of a / B minus a so this is f of 4us f 2 over 4 - 2 so let's calculate F of 4 let's replace x with 4 so it's going to be 4^ 2 + 2 * 4 - 3 and if we plug in two it's 2^ 2 + 2 * 2 - 3 ID 4 - 2 which is two so 4^ 2 is 16 2 * 4 is 8 - 3 2^ 2 is 4 but don't forget to distribute the negative sign 2 * 2 is 4 but if you add the negative is-4 and then * -3 that's + 3 / 2 so notice that the 3 is cancel -3 + 3 is z and 8 cancels with4 and4 so it's 16 / 2 which is 8 so this is the average rate of change of the interval between 2 and four now this represents the slope not of the tangent line but the slope of the secant line so now let's calculate the instantaneous rate of change at xal 3 so to do that we need to find frime of 3 the instantaneous rate of change is the slope of the tangent line at that point but remember the average rate of change is the slope of the secant line so let's find the derivative the derivative of x^2 is 2x to the first Power which is simply 2x the derivative of 2x is 2 and the derivative of a constant like -3 is z so frime of X is 2x + 2 frime of 3 is going to be 2 * 3 + 2 which turns out to be exactly eight so this problem worked out nicely the average rate of change turned out to be the instantaneous rate of change let's try another example let's say that f ofx is equal to x 4 + 5x what is the average rate of change or the slope of the secet line in this interval from 1 to four so let's use the equation F of B minus F of a / B minus a so that's going to be F of 4 - F of 1 / 4 - 1 F of 4 4 base to the 4 power 4 * 4 is 16 * 4 is 64 * 4 is 256 and we have to plug in four into 5x so 5 * 4 is 20 and F of 1 1 to 4th power is 1 5 * 1 is 5 and on the bottom 4 - 1 is 3 so we have 256 + 20 which is 276 minus 6 that's 270 / 3 which is 90 so the slope of the secant line is 90 that's the average rate of change now how can we estimate the instantaneous rate of change or the slope of the tangent line at let's say two how can we do that well first let's find the exact answer let's find frime of X what is the derivative of x to the 4th power this is going to be 4X Cub the derivative of 5x is 5 so now if we plug in two it's going to be 2 3 which is 8 * 4 that's 32 + 5 which is 37 so that's the easiest way to find the slope of the tangent line at xal 2 or the instantaneous rate of change at that point but now let's estimate that answer using the average rate of change equation so let's estimate frime of 2 let's choose an interval between 1 and three if you want to estimate the instantaneous rate of change at two you want to pick two numbers where two is the middle two is the midpoint of 1 and three so let's see what F of 3 minus F of 1 / 3 minus 1 is equivalent to now keep in mind as the difference as Delta X or the value of the denominator as it becomes very close to zero the average rate of change approaches the instantaneous rate of change so since one is pretty much far from three our answer shouldn't be very accurate but we're going to decrease the value of the denominator to approach the instantaneous rate of change which we know it's to be 37 so what is f of three so 3 4th power + 5 * 3 and then minus F of 1 which is 1 4th power + 5 * 1 / 2 3 to 4th is 81 that's 3 * 3 * 3 * 3 5 * 3 is 15 81 + 15 is 96 1 and 5 is 6 96 - 6 is 90 / 2 this is 45 which is not very close to 37 so let's change the interval let's make it 1.9 and 2.1 so it's going to be F of 2.1 minus F of 1.9 / 2.1 - 1.9 so two is still in between 1.9 and 2.1 but the difference now is not two it's. 2 so the answer should be closer to 37 but let's see what it is so we have 2.1 raised to 4th power + 5 * 2.1 that's F of 2.1 minus F of 1.9 which is 1.9 to 4th power + 5 * 1.9 and 2.1 - 1. 9 is.2 so 2.1 to 4th power if you type it in that's 19448 1 + 5 * 2.1 so that's going to be 29.94 81 minus 1.9 to 4th is 13.03 21 + 5 * 1.9 so that's going to be 22. 5321 ID 0.2 29.94 H1 minus 22.5 321 that's 7416 ID 0. 2 this is about 37.8 so right now this is close enough to the instantaneous rate of change at two which is exactly 37 so as you can see you can approximate the instantaneous rate of change using the average rate of change now you might be wondering why do I need to know this I mean can I just find the derivative using the easy method get the answer and then I'm done well the answer is yes if you have the function you can always do it the easy way you don't have to do it this way but sometimes you got to find the instant mous rate of change and you don't have the function let's say if they give you a graph or a table of values and there's no function there's no F ofx then you have to do it this way so that's why it's important to be able to find the instantaneous rate of change using the average rate of change in the appropriate way now let's work on another example let's say f ofx is x^2 + 3x x - 5 so what I want you to do is find the derivative and find the slope of the tangent line at x = 3 and then use limits to find the derivative as a function and also the slope of the tangent line at xal 3 so first let's do it the easy way frime of X is going to be equal to the derivative of x^2 is 2X for 3x is three and for a constant is zero so that's the derivative as a function to find the slope of the tangent line or the instantaneous rate of change plug in three 2 * 3 + 3 is equal to 9 so now use limits to get this answer the derivative as a function and the slope of the tangent line so to get the derivative as a function it's going to going to be the limit as H approaches Z FX + H - FX / H so it's going to be x + h^ 2 + 3 * x + H - 5 so that's F ofx + H minus F ofx which is simply the original function divid H so how can we simplify this expression what would you do we need to foil x + h^2 and let's not forget to rewrite the limit so we have the limit as H approaches z x + H * x + H and then let's distribute the 3 to X+ H so it's going to be 3x + 3 h - 5 and then we'll have to distribute the negative sign so it's x^2 - 3x + 5 / H so 3X and- 3X will cancel and five and Nega five will cancel as well so now we have the limit as H approaches zero x + H * x + H so it's a foil it's going to be x * X which is x^2 and then x * H and H * X which is xh + xh or 2 xh and then H * h which is h^2 and then we have 3 H - x^2 / H so now we can cancel X2 and from what remains let's factor out an H so we have the limit as H approaches z h * 2x + H + 3 now we can cancel h H so this is going to disappear now we have the limit as H approaches Z 2x + H + 3 at this point we can replace H with zero so we get 2x + 3 which is frime of X that's the answer that we had before so that's how you could find the derivative of the function at any value of x now since frime of X was 2x + 3 when we plug in three we saw that the the slope of the tangent line was equal to 9 so let's see if we can get n using limits so if you want to find the slope of the tangent line if you want to get a number instead of a function that depends on X use this form of the equation it's the limit as X approaches a f ofx minus F of a / x - A so a is three in this example so frime of 3 which we know to be N9 is going to be the limit as X approaches three F of three I mean f ofx minus F of 3 / x - 3 f ofx is going to be x^2 + 3x x - 5 so f ofx is simply the original equation and F of 3 we need to plug in three so it's going to be 3^ 2 + 3 * 3 - 5 all divid x - 3 so what we now have is x^2 + 3x - 5 3^ 2 is 9 3 * 3 is 9 and let's distribute the negative sign and then * 5 is POS 5 so at this point we can cancel five so what we have left over the limit as X approaches 3 x 2 + 3x -9 - 9 is 18 / x - 3 right now we can't plug in three because we will get a zero in the bottom so we need to factor the numerator so we have a trinomial With The Lean coefficient of one what two numbers multiply to8 and add to three this is going to be positive 6 and-3 6 * -3 is8 but 6 + -3 is pos3 so we could Factor the expression like this it's going to be x + 6 * x - 3 / x - 3 if the leading coefficient is 1 you could simply write x + and x - 3 so we can see that the x - 3 Expressions will cancel so once we get rid of that factor all we have left over is the limit as X approaches 3 x + 6 so now we can replace x with three so it's going to be 3 + 6 which is 9 which is the slope of the tangent line at x equal string so that's how you can use limits to find the slope of the tangent line and now you know how to use limits to find the derivative of the function and you also know how to do it easily so now let's work on some examples where we have a table of values so using this table what is the average rate of change on the interval 1 to three what would you do to find it so the average rate of change is f of 3 minus F of 1 / 3 - 1 so it's going to be 11.6 - 7.1 / 2 11.6 - 7.1 that's 4.5 half of 4.5 is about 2.25 so that's the average rate of change on the interval from 1 to three now let's say if you get a question that ask you what is the slope of the secut line between the points 2 and four so to find the slope of the SEC line is the same as finding the average rator change it's F of 4 minus F of 2 / 4 - 2 F of 4 is 13.3 F of 2 is 8.9 so 13.3 minus 8.9 that's 4.4 if you divide it by two the average rate of change or the slope of the secant line is 2.2 now what would you do if you wanted to find the instantaneous rate of change at four so how can you find fime of four to find frime of 4 we can't do it directly all we can do is approximate it using the average rate of change in the interval 3 to 5 notice that four is the midpoint of three and five so it's going to be F of 5us f of 3 / 5 - 3 F of 5 is 16.1 F of 3 is 11.6 / 2 16.1 - 11.6 is 4.5 ID 2 so the instantaneous rate of change is approximately 2.2 5 it's not an exact answer it's simply an approximation now how can you approximate the slope of the tangent line at 1.5 this is the same as the instantaneous rate of change at 1.5 so we need to pick two numbers where 1.5 is in the middle of those two numbers we can use zero and three or better yet 1 and two 1 and two is Clos to 1.5 than 0 and 3 but let's do it both ways let's see what the answer is for 0 and 3 and then 1 and two so F of 3 - F of 0 / 3 - 0 f of 3 is about 11.6 f of0 is 5 so 11.6 minus 5 / 3 is about 2.2 now let's see what happens between one and two one and two is going to give us a better estimation so F of 2 - F of 1 / 2 - 1 so it's 8.9 - 7.1 / by one which is about 1.8 so for this particular function 1.8 is going to be a better estimate of the instantaneous rate of changeed any the other answer since one and two is closer to 1.5 so if you're given a data table or even a graph the best way to estimate the instantaneous rate of change or the slope of this T line is to use the aage trative change formula and try to get as close close as possible to this number and make sure this number is in between A and B so that is it for this video thanks for watching and have a great day