Stony Brook MAT 125 Spring 2015
Intuitive explanation of the Chain Rule

Start   a chain rule which is what we need to do when we have a function described as the composition of two functions each of which we know how to take the derivative of so for example maybe I have the sin of 3x squared plus 2 and I want to think of this as one function with another function inside it so here
0:34f u is the sign u and g of x is 3x squared plus 2 which is u when i put them together I want to somehow relate the derivative here i take the derivative of the sine with respect to u and that will be the
1:00cosine and if i take the derivative of 3x squared with respect X that will be 6x and I want to somehow put these together and really what the chain rule says is that these two derivatives just multiply but we need to so the derivative will be just of the sin 3x plus two will be just this
1:34derivative times that but I made up u so i have to rewrite you in terms of X and us 3x squared plus 2 so this is really derivative I want 6x times usually write the 6x in front so
2:06it's the derivative of the outer function times the derivative of the inner function why would this be so so let's think about it's not just an arbitrary rule that i wrote down i'm gonna erase this example and why would we expect this to work this way all so we want we want to take the derivative of the composition
2:33like this and what is it so let's think about what the derivative is in the first place i have let's just think about what does G prime of X represent well I think of g is some kind of a transformational object i put in number like X and outcomes another number 3 x square to that's what she does it transforms an X into a 3x
3:06squared plus 2 and what the derivative is telling me is that if i move ex a little bit in one direction or another then the amount that comes out will be moved by about 6 x the derivative multiplies my input error and ads on about of wiggle that we x 66 x times
3:40let's put this wiggle as H exact times H so it tells me that my wiggles on the input are going to be magnified by a fat after of 6x on the output ok so now what is the derivative of F do well the derivative of F says let's call
4:03this number you said that if I take a number you and I feed it into the f machine that's not like dropping the f-bomb that's a different thing I feed you into the f machine outcomes sign of you but the derivative tells me that any amount of wiggling that i have on you or any small amount of whittling that i have on you will be magnified by a factor of the cosine so if i have a so now if i hook
4:38these two things together then we see that the amount of Wiggles are going to multiply that is if i take an x and move it a little bit and then I feed it into G then that will also move a little bit and that little bit will be controlled exactly by the derivative 6x and then if i take that output
5:01and move it and put it into the f machine but it's off a little bit because I moved it it will come out factored by the cosine so that means that this derivative will be 6x which is what I get out of you out of removing the X by the g machine times the amount that i get by moving f but you was of course 3x squared plus 2
5:36more formally we can write this by saying that the derivative of this composition is the derivative of the outside function plug-in where it is times the derivative of the inside if we write this in the other
6:02notation in the likeness notation here we're on thinking you is g of x and y is f of g of x f you then the amount that why changes when i move ex a little bit is this same as the amount that y changes when I move x a little bit
6:40that's F prime u times the amount that you change when i move x a little bit works just like fractions i can just cancel the du's this is strictly speaking not true but it works that way if we do the proof which i'm not doing
7:04in this and you'll see that actually we can carefully adjust these so that they they do cancel so it works this is the chain rule as but so we have the chain rule here and why it works