# 2.4: Derivatives of Trigonometric Functions

## Learning Objectives

A student will be able to:

- Compute the derivatives of various trigonometric functions.

If the angle is measured in radians,

and

We can use these limits to find an expression for the derivative of the six trigonometric functions and . We first consider the problem of differentiating , using the definition of the derivative.

Since

The derivative becomes

Therefore,

It will be left as an exercise to prove that

The derivatives of the remaining trigonometric functions are shown in the table below.

**Derivatives of Trigonometric Functions**

Keep in mind that for all the derivative formulas for the trigonometric functions, the argument is measured in radians.

**Example 1:**

Show that

*Solution:*

It is possible to prove this relation by the definition of the derivative. However, we use a simpler method.

Since

then

**Example 2:**

Find .

*Solution:*

Using the product rule and the formulas above, we obtain

**Example 3:**

Find if . What is the slope of the tangent line at ?

*Solution:*

Using the quotient rule and the formulas above, we obtain

To calculate the slope of the tangent line, we simply substitute :

We finally get the slope to be approximately

**Example 4:**

If , find .

*Solution:*

Substituting for ,

Thus .

## Multimedia Links

For examples of finding the derivatives of trigonometric functions **(4.4)**, see Math Video Tutorials by James Sousa, The Derivative of Sine and Cosine (9:21).

## Review Questions

Find the derivative of the following functions:

- If , find
- Use the definition of the derivative to prove that

## Review Answers