Special Derivatives

We will examine the derivative of the following functions:

ln(x)
ex

The derivative of the functions is another application of the chain rule:

h’(x) = g’(x) * f’(g(x))

The derivative of ln(x) is as follows:

derivatives 1

u is the expression inside the natural log function.

Example 1 – Differentiate f(x) = ln (x3 + 2x + 1).

u = x3 + 2x + 1, u’ = (3)*x3-1 + (1)*2x1-1 + 0 = 3x2 + 2

Applying the chain rule,

derivatives 2

Example 2 – Differentiate f(x) = ln(sin(x)).

u = sin(x), u’ = cos(x)

Applying the chain rule,

derivatives 3

The derivative of ex is as follows:

derivatives 4

u is the expression inside the exponent of the function.

Example 1 – Differentiate f(x) = e2x+5.

u = 2x + 5
u’ = 2x1-1 + 0 = 2x0 = 2

Applying the chain rule,

f’(x) = 2e2x+5

Example 2 – Differentiate f(x) = e-cos(3x)

u = -cos(3x)
u’ = 3sin(3x)

Applying the chain rule,

f’(x) = 3sin(3x)*e-cos(3x)