**Introduction to derivative of sin 4x:**

In calculus, the **derivative** is a measure of how a function changes as its input changes. The process of finding a derivative is called differentiation. Differentiation is a
method to compute the rate at which a dependent output y changes with respect to the change in the independent input x. The derivative of y with respect to x is given by `(dy)/(dx)` .

Here, we are going to learn derivative of sin 4x and its related functions from the following example and practice problems.(Source: Wikipedia)

**Example 1:**

**Find the derivative of y = sin 4x.**

**Solution:**

Step 1: Given function

y = sin 4x

Step 2: Differentiate the given function y = sin 4x with respect to ' x '

`(dy)/(dx)` = cos 4x (4)

= 4 cos 4x

**Example 2:**

**Find the derivative of y = sin ^{2} 4x.**

**Solution:**

Step 1: Given function

y = sin^{2} 4x

Step 2: Using trigonometric formula, we can write the given function as follows.

y = `(1 - cos 8x)/2`

y = `1/2 ` - `(cos 8x)/2`

Step 2: Differentiate the above function with respect to ' x '

`(dy)/(dx)` = 0 - `1/2` (- sin 8x) (8)

=+ 4 sin 8x

**Example 3:**

**Find the derivative of y = sin 2x sin 4x.**

**Solution:**

** ** Step 1: Given function

y
= sin 2x sin 4x

Step 2: Use product - to - sum formula and write the given function as follows.

y = sin 2x sin 4x

y = `1/2` {cos (2x - 4x) - cos (2x + 4x)}

[sin A sin B = cos (A - B) - cos (A + B)]

y = `1/2` {cos (-2x) - cos 6x}

y = `1/2` {cos (2x) - cos 6x}

y = `1/2` cos 2x - `1/2` cos 6x

Step 3: Differentiate each term with respect to ' x '.

`(dy)/(dx)` = `1/2` (- sin 2x) (2) - `1/2` (- sin 6x) (6)

= - sin 2x + 3 sin 6x

1) Find the derivative of y = - 7 sin 4x

2) Find the derivative of y = (- 6)sin (-4x)

3) Find the derivative of sin 4x cos 2x

**Solutions:**

1) - 28 cos 4x

2) 24 cos 4x

3) 3 cos 6x + cos 2x