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# How to use the chain rule with trigonometric functions

## Derivatives of Trigonometric Functions

In the previous article, we discussed the derivative of the trigonometric function expressed by the following formulas,

 f(x) f’(x) sin x cos x cos x -sin x

By using chain theorems, we can develop derivatives of complex trigonometric functions, such as the following examples,

Example 1
Find the result of the function derivative y = F(x) = sin 5x.
For example, y = sin u and u = 5x then

Example 2.
Find the result of the function derivative y = F’(x) = cos (2x – 8)

Example 3.
Find the result of the function derivative y = y = F(x) = sin3 x

## Integral Trigonometry Function

Given that integrals are anti-differential, we can determine the basic formulas of the following trigonometric functions,
∫ sin x dx = - cos x + c
∫ cos x dx = sin x + c

Example 4.

find the integral of a function ∫ 4 sin x dx using the chain rule.
∫ 4 sin x dx = - 4 cos x + c

Example 5.

find the integral of a function
using the chain rule.