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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:

Apply the formula above to obtain
$f '(x) = \dfrac{1}{x \ln 3}$

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 $f$
$f '(x) = \dfrac{1}{x} + 12x$

Example 3

Find the derivative of

$f(x) = \dfrac{\log_3 x}{1 - x}$
Solution to Example 3:

Let $g(x) = \log_3 x$ and $h(x) = 1 - x$ , function $f$ is the quotient of functions $g$ and $h$ : $f(x) = \dfrac{g(x)}{h(x)}$ . Hence we use the quotient rule, $f '(x) = \dfrac{(h(x) g '(x) - g(x) h '(x))}{(h(x))^2}$ , to find the derivative of function $f$ .
$g '(x) = \dfrac{1}{(x \ln 3)}$
$h '(x) = -1$
$f '(x) = \dfrac{(1 - x)(\dfrac{1}{(x \ln 3)}) - (\log_3 x)(-1)}{(1 - x)^2}$

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.
$f '(x) = \dfrac{dy}{du} \cdot \dfrac{du}{dx}$
$\dfrac{dy}{du} = \dfrac{1}{u}$ and $\dfrac{du}{dx} = -4$
$f '(x) = \dfrac{1}{u}(-4) = \dfrac{-4}{u}$
Substitute $u = -4x + 1$ in $f '(x)$ above
$f '(x) = \dfrac{-4}{(-4x + 1)}$

Exercises

Find the derivative of each function.
1)

$f(x) = \ln(x^2)$
2)

$g(x) = \ln x - x^7$
3)

$h(x) = \dfrac{\ln x}{(2x - 3)}$
4)

$j(x) = \ln (x + 3) \ln (x - 1)$

Solutions to the Above Exercises

$f '(x) = \dfrac{2}{x}$
2)

$g '(x) = \dfrac{1}{x} - 7x^6$
3)

$h '(x) = \dfrac{(2x - 3 - 2x \ln x)}{x(2x -3)^2}$
4)

$j '(x) = \dfrac{\ln (x + 3)}{x - 1} + \dfrac{\ln (x - 1)}{x + 3}$

Differentiation of Logarithmic Functions

First Derivative of a Logarithmic Function to any Base

Examples with Solutions

Example 1

Example 2

Example 3

Example 4

Exercises

Solutions to the Above Exercises

More References and links