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 problems on derivative of sin 4x:

 

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 = sin2 4x.

Solution:

Step 1: Given function

y = sin2 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

 

 

Practice problems on derivative of sin 4x:

 

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