 Let's briefly recap the major ideas from section 2.5 of active calculus on the chain rule. The chain rule is a derivative rule that we use when we're differentiating a composite function. That is, a function in which the input passes through two or more steps of calculations like links in a chain. An example of a composite function would be y equals the square root of x cubed minus 2. Now if I were to evaluate this function at say x equals 3, this would involve two distinct steps that have to be done one after the other. First I'd have to evaluate x cubed minus 2 at x equals 3, and that would give me 25. And then I would take the result of that calculation and find its square root next to get 5. Again, notice that although y is really just one function, in reality there are two functions that make it up. And when we pass an input through, then the order in which the links in the chain operate on this input is very important. Back in high school algebra courses, we learned that in notational language, we would look at this composite function and let g of x equal x cubed minus 2, the first link in the chain. And then if we let f be the function that takes the square root, then y is equal to f of g of x. That's the function f that computes a square root with the function g of x plugged into it. With this basic setup, we can state the chain rule. If g is differentiable at x, and f is differentiable at g of x, then the composite function c, that's defined by c of x equals f of g of x, is also differentiable at x, and c prime of x is equal to the following. f prime of g of x times g prime of x. So in English, we are saying that the derivative of this composite function c of x, which is f of g of x, where g of x is the inside, quote unquote, function or the first link in the chain. When f is the outside function of the second link in the chain, then the derivative of c is f prime evaluated at the original g of x, all times the derivative of g of x. So the chain rule is perhaps the most important rule we've learned so far for derivative taking because composite functions show up so often in applications. So we will now do a number of examples to get the gist of how the chain rule works.