 Hello, and welcome to a screencast about the derivatives of composite functions, and specifically, polynomials. Okay, so whenever you have a composite function, that means you can identify an inside function and an outside function. That means you're going to want to use the chain rule. So the chain rule says that if g is differentiable at x, and f is differentiable at g of x, then the composite function c of x is f of g of x is also differentiable at x, and the derivative is defined to be f prime of g of x times g prime of x. Okay, so basically you need to identify an inside function and an outside function. So decide which function is being composed into the other one. Then once you can identify those, then putting together the chain rule is fairly straightforward. And usually what the chain rule allows you to do is take derivatives of ugly functions, like for example y equals x squared minus one, all of that quantity cubed. Okay, that's fairly ugly. Take that, break it down into smaller pieces, and then those pieces are a lot easier to do the derivative of. Okay, so our concept check here. I want you to pause the video and think about this. Using the above definition of c of x. Okay, so that means that the outside functions f and the inside functions g. So g is being composed into f. Which function should we use for f and g in the chain rule? So I have four options here. One where f of x is x squared minus one, g of x is x cubed. Another one where those are switched around a little bit. f of x is x squared, g of x is x cubed minus one. C, f of x is x cubed, g of x is x squared minus one. Or D, f of x is x cubed minus one, and g of x is x squared. So pause the video for just a second, and then come back with your choice. Okay, so because g of x is being composed into f, that means g is technically your inside function. Your inside function here is x squared minus one. So the option that's correct is going to be c. So your outside function is f of x is x cubed, your inside function is g of x is x squared minus one. Okay, so these functions are fairly straightforward to do, like I said for the derivatives. So if f is x cubed, your outside function, g is x squared minus one, that's your inside function. Then the derivative of f using the power rule is 3x squared. The derivative of g, your inside function, again using the power rule, is simply 2x. Okay, so for this one, you know again, x squared, drop dot two down, reduce your power by one, and then the minus one here goes away because the derivative of that's just zero. Okay, so the power or the chain rule here says that you're going to take your inside function, it's going to stay the same, but that's going to now be put into your derivative. Okay, so to kind of give you a little bit of a visual here, and this may help some of you, it may not help others, you're going to take this function, that's your inside, and you're going to put it into the derivative. Okay, then we're going to multiply by the derivative of the inside, which is 2x. Okay, so how will those pieces go together then? It's just following the chain rule up here. So our function was defined to be y, so we'll say dy dx is going to equal three times, so now instead of just writing an x, I'm going to have to replace that with my function g. So that's going to be x squared minus one, and all of that's to the second power, and then times the derivative of that inside function, 2x, and that is a good example of the chain rule using a polynomial. Thank you for watching.