What I hope to do in this video is give you a satisfying proof of the product rule. So let's just start with our definition of a derivative. So if I have the function f of x, and if I wanted to take the derivative of it, by definition, by definition, the derivative of f of x is the limit as h approaches zero of f of x plus h minus f of x, all of that over, all of that over h. If we want to think of it visually, this is the slope of the tangent line and all of that. But now I want to do something a little bit more interesting. I want to find the derivative with respect to x, not just of f of x, but the product of two functions, f of x times g of x. And if I can come up with a simple thing for this, that essentially is the product rule. Well, if we just apply the definition of a derivative, that means I'm going to take the limit as h approaches zero.