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.
Answer:
For example, y = sin u and u = 5x then


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



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



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.
Answer:
∫ 4 sin x dx = - 4 cos x + c

Example 5.

find the integral of a function 
using the chain rule.
Answer:



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