# Finding the Derivative of sec^2(x)

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Lesson Transcript
Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

In this lesson, we'll see how to find the derivative of sec^2(x). We'll go over the chain rule and a few well-known formulas that will help us find this derivative and see how integrals can be used to check the answer.

## Steps to Solve

In order to find the derivative of sec 2 x, we need to use the chain rule, along with these two well-known derivative formulas:

The chain rule is a formula we use to calculate the derivative of a composition of functions, where a composition of functions is a function within a function.

In our problem, sec 2 x can also be looked at as (sec x) 2. This is the same as plugging g(x) = sec x into the function f(x) = x 2 for x. That is, f(g(x)) = (sec x) 2 = sec 2 x, making it a composition of functions, or a function within a function.

The chain rule in symbol form is as follows.

We see that the chain rule states that if we're taking the derivative f(g(x)), we take the derivative of the function f and then plug g into the result. We then multiply by the derivative of the function g. Therefore, to find the derivative of sec 2 x, we take the following steps:

1. Find the derivative of f(x) = x 2, which is f'(x) = 2x.
2. Plug g(x) = sec x into f'(x) from step 1.
3. Multiply the result of step 2 by the derivative of g(x) = sec x, which is g'(x) = secx tanx.

Let's get started! As shown in the image currently below, in step 1 we found the derivative of f(x) = x 2, which is f'(x) = 2x. Now we find f'(g(x)). That is, we plug g(x) = sec x into f'(x), as shown here:

The next thing we need to do is multiply the result, f'(g(x) = 2sec x, by the derivative of g(x) = sec x. We know the derivative of g(x) = sec x is g'(x) = secx tanx, so we multiply 2sec x by secx tanx to get our answer.

The following shows what we did in a nice organized manner:

We see that the derivative of sec 2 x is 2sec 2 x tan x.

## Checking Your Work

To make sure that we've calculated the derivative of something correctly, we can check it using integrals. An integral is an anti-derivative, so it undoes the process of taking the derivative of something. We have the following rule relating integrals and derivatives:

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