The product rule
What we will talk about in this video is the product rule, which is one of the fundamental ways of evaluating derivatives. And we won't prove it in this video, but we will learn how to apply it. And all it tells us is that if we have a function that can be expressed as a product of two functions-- so let's say it can be expressed as f of x times g of x-- and we want to take the derivative of this function, that it's going to be equal to the derivative of one of these functions, f prime of x-- let's say the derivative of the first one times the second function plus the first function, not taking its derivative, times the derivative of the second function. So here we have two terms. In each term, we took the derivative of one of the functions and not the other, and we multiplied the derivative of the first function times the second function plus just the first function times the derivative of the second function. Now let's see if we can actually apply this to actually find the derivative of something. So let's say we are dealing with-- I don't know-- let's say we're dealing with x squared times cosine of x. Or let's say-- well, yeah, sure. Let's do x squared times sine of x. Could have done it either way. And we are curious about taking the derivative of this. We are curious about what its derivative is. Well, we might immediately recognize that this is the product of-- this can be expressed as a product of two functions. We could set f of x is equal to x squared, so that is f of x right over there. And we could set g of x to be equal to sine of x. And there we have it. We have our f of x times g of x. And we could think about what these individual derivatives are. The derivative of f of x is just going to be equal to 2x by the power rule, and the derivative of g of x is just the derivative of sine of x, and we covered this when we just talked about common derivatives. Derivative of sine of x is cosine of x. And so now we're ready to apply the product rule. This is going to be equal to f prime of x times g of x. So f prime of x-- the derivative of f is 2x times g of x, which is sine of x plus just our function f, which is x squared times the derivative of g, times cosine of x. And we're done. We just applied the product rule.
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