Derivative of Sin 3x

 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 3x and its related functions from the following example and practice problems.

                                                                                                                                                                                (Source: Wikipedia)

 

Example problems on derivative of sin 3x:

 

Example 1:

Find the derivative of y = sin 3x.

Solution:

      Step 1: Given function

                                y = sin 3x

      Step 2: Differentiate the given function y = sin 3x with respect to ' x '

                                `(dy)/(dx)` = cos 3x (3)

                                             = 3 cos 3x

Example 2:

Find the derivative of y = 5 sin 3x.

Solution:

      Step 1: Given function

                                y = 5 sin 3x

      Step 2: Differentiate the given function y = 5 sin 3x with respect to ' x '

                                `(dy)/(dx)` =  5 cos 3x (3)

                                             = 15 cos 3x

 

Example 3:

Find the derivative of y = sin 3x cos 5x.

Solution:

      Step 1: Given function

                                y = sin 3x cos 5x

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

                                y = sin 3x cos 5x

                                y = `1/2` {sin (3x + 5x) + sin (3x - 5x)}         [sin A cos B = sin (A + B) + sin (A - B)]

                                y = `1/2` {sin 8x + sin (- 2x)}

                                y = `1/2` {sin 8x - sin 2x}

                                y = `1/2` sin 8x - 1/2 sin 2x

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

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

                                             = 4 cos 8x - cos 2x

 

Practice problems on derivative of sin 3x:

 

1) Find the derivative of y = 15 sin 3x

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

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

Solutions:

1) 45 cos 3x

2) 5 cos 10x - sin 2x

3) - 16 cos 4x