# Differentiation of Logarithmic Functions

Examples of the derivatives of logarithmic functions, in calculus, are presented. Several examples, with detailed solutions, involving products, sums and quotients of exponential functions are examined.

## First Derivative of a Logarithmic Function to any Base

The first derivative of f(x) = log_{ b} x is given by

** f '(x) = 1 / (x ln b) **

Note: if f(x) = ln x , then f '(x) = 1 / x

## Examples

### Example 1

Find the derivative of f(x) = log_{ 3} x

__Solution to Example 1:__

- Apply the formula above to obtain

f '(x) = 1 / (x ln 3)

### Example 2

Find the derivative of f(x) = ln x + 6x^{ 2}

__Solution to Example 2:__

- Let g(x) = ln x and h(x) = 6x
^{ 2}, function f is the sum of functions g and h: f(x) = g(x) + h(x). Use the sum rule, f '(x) = g '(x) + h '(x), to find the derivative of function f

f '(x) = 1 / x + 12x

### Example 3

Find the derivative of f(x) = log_{ 3} x / ( 1 - x )

__Solution to Example 3:__

- Let g(x) = log
_{ 3} x and h(x) = 1 - x, function f is the quotient of functions g and h: f(x) = g(x) / h(x). Hence we use the quotient rule, f '(x) = [ h(x) g '(x) - g(x) h '(x) ] / h(x)^{ 2}, to find the derivative of function f.

g '(x) = 1 / (x ln 3)

h '(x) = -1

f '(x) = [ h(x) g '(x) - g(x) h '(x) ] / h(x)^{ 2}

= [ (1 - x)(1 / (x ln 3)) - (log_{ 3} x)(-1) ] / (1 - x)^{ 2}

### Example 4

Find the derivative of f(x) = ln (-4x + 1)

__Solution to Example 4:__

- Let u = -4x + 1 and y = ln u, Use the chain rule to find the derivative of function f as follows.

f '(x) = (dy / du) (du / dx)

- dy / du = 1 / u and du / dx = -4

f '(x) = (1 / u)(-4) = -4 / u

- Substitute u = -4x + 1 in f '(x) above

f '(x) = -4 / (-4x + 1)

## Exercises

Find the derivative of each function.

1) f(x) = ln(x^{ 2})

2) g(x) = ln x - x^{ 7}

3) h(x) = ln x / (2x - 3)

4) j(x) = ln (x + 3) ln (x - 1)
### Solutions to the Above Exercises

1) f '(x) = 2 / x

2) g '(x) = 1 / x -7x^{ 6}

3) h '(x) = (2x - 3 - 2x ln x) / [ x(2x -3)^{ 2} ]

4) j '(x) = ln (x + 3) / (x - 1) + ln (x - 1) / (x + 3)
### More References and links

differentiation and derivatives