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# Quotient rule from product & chain rules

Sal shows how you can derive the quotient rule using the product rule and the chain rule (one less rule to memorize!). Created by Sal Khan.

Video transcript

We already know that the
product rule tells us that if we have the product of
two functions-- so let's say f of x and g of x--
and we want to take the derivative of this
business, that this is just going to be equal
to the derivative of the first function,
f prime of x, times the second function, times g
of x, plus the first function, so not even taking its
derivative, so plus f of x times the derivative
of the second function. So two terms, in each term
we take the derivative of one of the functions and not the
other, and then we switch. So over here is the
derivative of f, not of g. Here it's the derivative
of g, not of f. This is hopefully a
little bit of review. This is the product rule. Now what we're
essentially going to do is reapply the
product rule to do what many of your calculus books
might call the quotient rule. I have mixed feelings
about the quotient rule. If you know it, it might make
some operations a little bit faster, but it really comes
straight out of the product rule. And I frankly always
forget the quotient rule, and I just rederive it
from the product rule. So let's see what
we're talking about. So let's imagine if we
had an expression that could be written as f
of x divided by g of x. And we want to take the
derivative of this business, the derivative of
f of x over g of x. The key realization
is to just recognize that this is the same thing
as the derivative of-- instead of writing f of
x over g of x, we could write this as f of x times
g of x to the negative 1 power. And now we can use
the product rule with a little bit
of the chain rule. What is this going
to be equal to? Well, we just use
the product rule. It's the derivative of the
first function right over here-- so it's going to
be f prime of x-- times just the second
function, which is just g of x to the negative 1
power plus the first function, which is just f of x,
times the derivative of the second function. And here we're going to have to
use a little bit of the chain rule. The derivative of
the outside, which we could kind of
view as something to the negative 1 power with
respect to that something, is going to be negative 1
times that something, which in this case is g of x
to the negative 2 power. And then we have to
take the derivative of the inside
function with respect to x, which is
just g prime of x. And there you have it. We have found the
derivative of this using the product rule
and the chain rule. Now, this is not
the form that you might see when
people are talking about the quotient
rule in your math book. So let's see if we can
simplify this a little bit. All of this is going to be equal
to-- we can write this term right over here as f
prime of x over g of x. And we could write
all of this as-- we could put this negative
sign out front. We have negative f of
x times g prime of x. And then all of that
over g of x squared. Let me write this a
little bit neater. All of that over g of x squared. And it still isn't in the
form that you typically see in your calculus book. To do that, we just have
to add these two fractions. So let's multiply the
numerator and the denominator here by g of x so that we have
everything in the form of g of x squared in the denominator. So if we multiply the
numerator by g of x, we'll get g of x right
over here and then the denominator will
be g of x squared. And now we're ready to add. And so we get the
derivative of f of x over g of x is equal to
the derivative of f of x times g of x minus-- not plus
anymore-- let me write it in white-- f of x
times g prime of x, all of that over g of x squared. So once again, you
can always derive this from the product rule
and the chain rule. Sometimes this might be
convenient to remember in order to work through some problems of
this form a little bit faster. And if you wanted to kind of see
the pattern between the product rule and the quotient
rule, the derivative of one function just times
the other function. And instead of
adding the derivative of the second function
times the first function, we now subtract it. And all that is over the
second function squared. Whatever was in the denominator,
it's all of that squared. So when we're taking
the derivative of the function in the
denominator up here, there's a subtraction, and then
we are also putting everything over the second
function squared.