Special Derivatives

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 (x³ + 2x + 1).

u = x³ + 2x + 1, u’ = (3)*x^3-1 + (1)*2x^1-1 + 0 = 3x² + 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 = 2x⁰ = 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)