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## The product rule

# Product rule

AP.CALC:

FUN‑3 (EU)

, FUN‑3.B (LO)

, FUN‑3.B.1 (EK)

## Video transcript

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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