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# Chain rule

AP.CALC:
FUN‑3 (EU)
,
FUN‑3.C (LO)
,
FUN‑3.C.1 (EK)

## Video transcript

what we're going to go over in this video is one of the core principles in calculus we're going to use it anytime you take the derivative of anything even reasonably complex and it's called the chain rule and when you're first exposed to it it can seem a little daunting and a little bit convoluted but as you see more and more examples it'll start to make sense and hopefully you'll even start to seem a little bit simple and intuitive over time so let's say that I had a function let's say I have a function H of X and it is equal to just for example I let's say it's equal to sine of X let's say it's equal to sine of x squared now I could have written that I could have written it like this sine squared of X but it'll be a little bit clearer using using that type of notation so let me make it so I have H of X and what I'm curious about is what is H prime of X so I want to know H prime of X which another way of writing it is there was the derivative of H with respect to X these are just different notations and to do this I'm going to use the chain rule I am going to use the chain rule and the chain rule comes into play every time any time your function can be used as a composition of more than one function and as that might not seem obvious right now but it will hopefully maybe by the end of this video or the next one now what I want to do is a little bit of a thought experiment a little bit of a thought experiment if I were to ask you what is the derivative with respect to X if I were just apply the derivative operator to x squared with respect to X what do I get well this gives me 2x we've seen that many many many many times now what if I were to take the derivative with respect to a of a squared what's the exact same thing I just swapped an A for the X's this is still going to be equal to 2ei now I will do something that might be a little bit more bizarre what if I were to take the derivative with respect to sine of X with respect to sine of X of of sine of X sine of x squared well wherever I had the X's up here the A's over here I just replace it with a sine of X so this is just going to be 2 times with the thing that I had so whatever I'm taking the derivative with respect to here was with respect to X here's respect to a here's with respect to sine of X so it's going to be 2 times sine of X now so the chain rule tells us that this derivative is going to be the derivative of our whole function with respect or the derivative of this outer function x squared the derivative of x squared the derivative of this outer function with respect to sine of X so that's going to be 2 sine of X 2 sine of X so we could view it as the derivative of the outer function with respect to the inner - sine of X we could just treat sine of X like it's kind of an X and it would have been just 2x but instead it's a sine of X we say 2 sine of X x times the derivative just in green times the derivative of sine of X with respect to x times the derivative of sine of X with respect to X well that's more straightforward a little bit more intuitive the derivative of sine of X with respect to X we've seen multiple times is cosine of X so times cosine of X and so there we've applied the chain rule it was the derivative of the outer function with respect to the inner so derivative of sine of x squared with respect to sine of X is 2 sine of X and then we multiply that times the derivative of sine of X with respect to X so let me make it clear this right over here is the derivative we're taking the derivative of we're taking the derivative of sine of x squared so let me make it clear that's what we're taking the derivative of with respect to sine of X with respect to sine of X and then we're multiplying that times the derivative of sine of X the derivative of sine of X with respect to with respect to X and this is where it start might start making a little bit of intuition you can't really treat these differentials this D whatever this DX this D sine of X as as a as a as a number and you really can't there's a notation makes it look like a fraction because intuitively that's what we're doing but if you were to treat them like fractions then you could think about canceling that and that and once again this isn't a rigorous thing to do but it can help with the intuition and then what you're left with is the derivative of this whole sine of x squared with respect to X so you're left with you're left with the derivative of essentially our original function sine of x squared with respect to X with respect to X which is exactly what dhdx is this right over here this right over here is our original function H that's our original function H so it might seem a little bit daunting now what I'll do in the next video is another several examples and then we'll try to abstract that this a little bit
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