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)
