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Mathematical Expression Editor

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Here we compute derivatives of products and quotients of functions

The product rule

Consider the product of two simple functions, say where and . An obvious guess for
the derivative of is the product of the derivatives:

Is this guess correct? We can check by rewriting and and doing the calculation in a
way that is known to work. Write with me

Hence so we see that So the derivative of is not as simple as . Never fear, we have a
rule for exactly this situation.

The product rule If and are differentiable, then

Let’s return to the example with which we started.

Let and . Compute:

Write with me

We could stop here, but we should show that expanding this out recovers our
previous result. Write with me

which is precisely what we obtained before.

Now that we are pros, let’s try one more example.

Compute:

Using the sum rule and the product rule, write with me

The quotient rule

We’d like to have a formula to compute This brings us to our next derivative
rule.

The quotient rule If and are differentiable, then

Compute:

Write with me

Compute:

Write with me

It is often possible to calculate derivatives in more than one way, as we have already
seen. Since every quotient can be written as a product, it is always possible
to use the product rule to compute the derivative, though it is not always
simpler.

Compute: in two ways. First using the quotient rule and then using the product rule.

First, we’ll compute the derivative using the quotient rule. Write with me

Second, we’ll compute the derivative using the product rule:

With a bit of algebra, both of these simplify to

Suppose we have two functions, , and , and we know that , , , and . What is the
slope of the tangent line to the curve at the point where ?

The slope of the tangent
line to the curve at is given by .

By the QuotientRule, this derivative is given by

Suppose we have two functions, , and , and we know that , , , and . What is the
slope of the tangent line to the curve at the point where ?

The slope of the tangent
line to the curve at is given by .

By the QuotientRule and the Product Rule, this derivative is given by