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  • 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.

What we will talk about in this video

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