Examples of the derivatives of logarithmic functions, in calculus, are presented. Several examples, with detailed solutions, involving products, sums, and quotients of logarithmic functions are examined.
First Derivative of a Logarithmic Function to any Base
The first derivative of \( f(x) = \log_b x \) is given by:
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:
\[ f'(x) = \dfrac{h(x)g'(x) - g(x)h'(x)}{[h(x)]^2} \]We find the individual derivatives:
\[ g'(x) = \dfrac{1}{x \ln 3} \] \[ h'(x) = -1 \]Substitute into the quotient rule formula:
\[ f'(x) = \dfrac{(1 - x)\left(\dfrac{1}{x \ln 3}\right) - (\log_3 x)(-1)}{(1 - x)^2} \]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 \( f \):
\[ f'(x) = \dfrac{dy}{du} \cdot \dfrac{du}{dx} \]Since \( \dfrac{dy}{du} = \dfrac{1}{u} \) and \( \dfrac{du}{dx} = -4 \):
\[ f'(x) = \dfrac{1}{u}(-4) = \dfrac{-4}{u} \]Substitute \( u = -4x + 1 \) back:
\[ f'(x) = \dfrac{-4}{-4x + 1} \]Exercises
Find the derivative of each function:
- \( f(x) = \ln(x^2) \)
- \( g(x) = \ln x - x^7 \)
- \( h(x) = \dfrac{\ln x}{2x - 3} \)
- \( j(x) = \ln(x + 3) \ln(x - 1) \)
Solutions to the Above Exercises
- \( f'(x) = \dfrac{2}{x} \)
- \( g'(x) = \dfrac{1}{x} - 7x^6 \)
- \( h'(x) = \dfrac{2x - 3 - 2x \ln x}{x(2x - 3)^2} \)
- \( j'(x) = \dfrac{\ln(x + 3)}{x - 1} + \dfrac{\ln(x - 1)}{x + 3} \)