Finding the Derivative of 1/cos(x)

Finding the Derivative of 1/cos(x)
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  • 0:00 Steps to Solve
  • 1:05 Solving
  • 2:21 Trigonometric Functions
  • 4:04 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

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

This lesson will explain how to find the derivative of 1 / cos(x). In the process of doing so, we will also look at the quotient rule for derivatives, derivatives of trigonometric functions, and some trigonometric identities.

Steps to Solve

We want to find the derivative of 1/cos(x). Notice that 1/cos(x) is a quotient. This tells us that we can use the quotient rule for derivatives to find this derivative. The quotient rule is shown on screen:


The Quotient Rule


In order to find the derivative of a quotient, we follow these steps.

1.) Identify the function in the numerator, f(x), and find its derivative, f'(x).

2.) Identify the function in the denominator, g(x), and find its derivative, g'(x).

3.) Plug these functions into the quotient rule and simplify.

Before we take our function through these steps, we are going to need to know a few key facts to get us through the solving process. You may already be familiar with these facts, but if not, here is a quick refresher.

  • 1/cos(x) = sec(x)
  • sin(x) / cos(x) = tan(x)
  • The derivative of a constant is zero.
  • The derivative of cos(x) is -sin(x).

Solving

Okay, let's get started.

Step 1: The first thing we want to do is identify the function in the numerator of 1/cos(x). We see this is 1, so we say f(x) = 1. We now want to find the derivative of this function. Since 1 is a constant, we know the derivative is 0 from our fact list. Therefore, f'(x) = 0.

Step 2: Our next step is to identify the function in the denominator of 1/cos(x). The function in the denominator is cos(x), so we let g(x) = cos(x). Now we find the derivative of cos(x), which our fact list says is -sin(x). Thus, g'(x) = -sin(x).

Step 3: Our last step is to plug f, g, f', and g' that we found in steps 1 and 2 into the quotient rule.


plugging in


Lastly, we simplify. To do this, we will use our two facts that 1/cos(x) = sec(x)

and that sin(x) / cos(x) = tan(x).


simplifying


We see that the derivative of 1/cos(x) is sec(x)tan(x)

Trigonometric Functions

As we just saw, being familiar with trigonometric identities and derivatives of trigonometric functions is essential when trying to find more complex derivatives involving trigonometric functions. Another reason to be familiar with these two concepts is because it can really reduce the amount of work needed to find a derivative. In our example, we can find the derivative of 1/cos(x) in two easy steps if we know some simple identities and derivatives of trigonometric functions.

Let's see how that's possible, but first, let's review the derivatives of trigonometric functions and some trigonometric identities. The derivatives of trigonometric functions are shown on screen:

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