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
\( f '(x) = \dfrac{1}{x \ln b} \)

Note: if \( f(x) = \ln x \), then \( f '(x) = \dfrac{1}{x} \)

Examples with Solutions

Example 1

Find the derivative of \( f(x) = \log_3 x \)
Solution to Example 1:

Example 2

Find the derivative of \( f(x) = \ln x + 6x^2 \)
Solution to Example 2:

Example 3

Find the derivative of \( f(x) = \dfrac{\log_3 x}{1 - x} \)
Solution to Example 3:

Example 4

Find the derivative of \( f(x) = \ln(-4x + 1) \)
Solution to Example 4:

Exercises

Find the derivative of each function.
1) \( f(x) = \ln(x^2) \)
2) \( g(x) = \ln x - x^7 \)
3) \( h(x) = \dfrac{\ln x}{(2x - 3)} \)
4) \( j(x) = \ln (x + 3) \ln (x - 1) \)

Solutions to the Above Exercises

1) \( f '(x) = \dfrac{2}{x} \)
2) \( g '(x) = \dfrac{1}{x} - 7x^6 \)
3) \( h '(x) = \dfrac{(2x - 3 - 2x \ln x)}{x(2x -3)^2} \)
4) \( j '(x) = \dfrac{\ln (x + 3)}{x - 1} + \dfrac{\ln (x - 1)}{x + 3} \)

More References and links

differentiation and derivatives