Learn about the quotient rule - when to use it and why

Say I give you a function which looks something like this:

and I ask you to take the derivative of it. What do you do?

You could for example serve me the solution on a plate, using the quotient rule:

The quotient rule

The quotient rule is just the product rule for a special case

This rule can be derived from the product rule in the following way:

We note that:

Using this fact, we write:

And that is the quotient rule again.

A small remark: in taking the derivative of we can use the chain rule: . The inner function is and the outer function is .

Example 1

The most fundamental example of the quotient rule is when taking the derivative of:

As mentioned above, we can use the chain rule to find the derivative. We can also use the quotient rule.

The numerator function would then be . The derivative of is zero, so:

This example has its own name: the reciprocal rule.

Example 2

Let and . Taking the derivative of the quotient, we get:

The numerator can be simplified with trigonometric rules, as . Thus, we get:

But, as we have seen when talking about trigonometric functions, . So we have just shown:

This can also be written as , if we do not use trigonometric identities when simplifying.

Good outline for calculus and short to-do list

We work hard to provide you with short, concise and educational knowledge. Contrary to what many books do.

Get exam problems for old calculus exams divided into chapters

The trick is to both learn the theory and practice on exam problems. We have categorized them to make it extra easy.

Apple logo
Google logo
© 2023 Elevri. All rights reserved.