The derivative of the sin and cosine functions is the application of the chain rule:

If h(x) =f(g(x)),

h’(x) = g’(x) * f’(g(x))

The derivative pattern is as follows:

u is the expression inside the trigonometric function.

**Example 1 – What is the derivative of h(x) = sin(5x)?**

u = 5x, u’=5

Applying the chain rule,

h'(x) = 5cos(5x)

**Example 2 – What is the derivative of h(x) = cos(9x)?**

u = 9x, u’ = 9

Applying the chain rule,

h'(x) = -9sin(9x)

**Example 3 – What is the derivative of h(x) = -3sin(4x)**

u = 4x, u’ = 4

Applying the chain rule,

h'(x) = -4*3cos(4x) = -12cos(4x)

**Example 4 – What is the derivative of h(x) = sin(x) * cos(x)**

Applying the product rule,

f(x) = sin(x)

f’(x) = cos(x)

g(x) = cos(x)

g’(x) = -sin(x)

h’(x) = f(x)*g’(x) + f’(x)*g(x) = sin(x)*-sin(x) + cos(x)*cos(x) = cos(2x)

**Example 5 – What is the derivative of h(x) = ?**

Applying the quotient rule,

f(x) = 4

f’(x) = 0

g(x) = cos(x)

g’(x) = -sin(x)