We will examine the derivative of the following functions:

ln(x)

e^{x}

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:

u is the expression inside the natural log function.

**Example 1 – Differentiate f(x) = ln (x ^{3} + 2x + 1).**

u = x^{3} + 2x + 1, u’ = (3)*x^{3-1} + (1)*2x^{1-1} + 0 = 3x^{2} + 2

Applying the chain rule,

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

u = sin(x), u’ = cos(x)

Applying the chain rule,

The derivative of e^{x} is as follows:

u is the expression inside the exponent of the function.

**Example 1 – Differentiate f(x) = e ^{2x+5}.**

u = 2x + 5

u’ = 2x^{1-1} + 0 = 2x^{0} = 2

Applying the chain rule,

f’(x) = 2e^{2x+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)}