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
\( f '(x) = \dfrac{-4}{(-4x + 1)} \)