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Video transcript

so we've got the function f of X is equal to two x to the third plus five x squared minus seven all of that to the eighth power and we want to find the derivative of our function f with respect to X now the key here is to realize that this function can be viewed as a composition of two functions how do we do that well let me diagram it out so let's say we want to start with I'll do it down here so I have some space so we're gonna start with an X and what's the first thing that we would do if you were just trying to evaluate it given some X well the first thing you would take 2 times that X to the third power plus 5 times that x squared and then minus 7 so what if we imagined a a function here that just did that first part that just evaluated X just evaluated to X to the third plus 5 x squared minus 7 for your X so let's call that the function U and so whatever you input into that function u you're gonna get 2 times that input to the third power plus 5 times that input to the second power minus 7 and so when you do that when you input with an X what do you output what do you output here well you're going to output U of X U of X which is equal to 2x to the third power plus 5 x squared minus 7 and I'll do it all in one color just so I don't have to keep changing colors so 2x to the third power plus 5 x squared minus 7 that is U of X now what's the next thing you're doing you're not done evaluating f of X yet you would then take that value and then input it into another function you would then take the 8th power of that value so then we will take that and input it into another function that let's call that function V and that function whatever input you give it and I'm using these squares is to say whatever input goes into that function you are going to take it to the 8th power and so in this case what do you get what do you end up with well you end up with V of U of X V of U of X or you could view this as V of 2x to the third to X to the third plus 5 x squared minus 7 or you could view this two x to the third plus five x squared minus seven all of that to the eighth power all of that to the eighth power and that's what f of X is so as we just saw f of X can be viewed as the composition of V and u this is f of X so if we write f of X if we write f of X being equal to V of U of X U of X then we see very clearly the chain rule is very useful here the chain rule tells us that F prime of X is going to be the derivative of V with respect to U so it's going to be V prime of not X but V prime of U of X the derivative of V with respect to U times the derivative of U with respect to X so u prime of X so we know we know a few things already so let's let me just write things down very clearly so we know that U of X is equal to 2x to the 3rd power plus 5x squared minus 7 what is U prime of X well here we're just going to use some derivative properties in the power rule 3 times 2 is 6 X 3 minus 1 is 2 6x squared 2 times 5 is 10 take one off that exponent it's going to be 10 X to the first power or just 10 X and the derivative of a constant is just zero so we can just ignore that so that's u prime of X now we know that V if we input an X into V so V of X would be equal to X to the 8th power V prime of X well we just use the power rule again that's 8x to the seventh power and so V prime of U of X so if you were to input U of X into V prime well it's going to be equal to it is going to be equal to 8 times U of X to the seventh power whatever you input into V Prime you're gonna mult you can take it to the seventh power and multiply it by 8 so you of X and that's the same thing that is the same day this is equal to eight times this entire expression U of X is 2x to the third power 2x to the third power plus five x squared minus seven so there you have it f prime of X it is equal to this which we just figured out V prime of U of X is all of this business so it's equal to eight eight times two x to the third plus five x squared minus seven all of that to the seventh power times u prime of X u prime of X we figured out is that so the times 6x squared plus plus 10x now as you get more practice with the chain rule you'll recognize this faster and actually you could do you won't have to write all of this down you'll be able to do a lot of it in your head you'll say okay I'm going to take the derivative of the outside function the blue function you could say with respect to what I have in the inside so if I was saying the derivative of X to the 8th it would be eight X to the seventh if that's with respect to X but if I'm taking the derivative of this with respect to the inside well where I had the X's before I would just have this U of X so it's going to be eight times this to the seventh power and I multiply that times the derivative of the inside which is 6x squared plus 10x