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Finding the Derivative of sec^2(x)

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:


deriv1


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.


The Chain Rule
deriv2


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:


deriv3


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.


deriv4


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


deriv5


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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