Derivatives of Sine and Cosine

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:

sine 1

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) = sine 2?

Applying the quotient rule,

f(x) = 4
f’(x) = 0
g(x) = cos(x)
g’(x) = -sin(x)

sine 3