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:

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 .

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.

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.