Derivative of Logarithmic Functions

Logarithmic functions are the inverse of exponential functions and play a crucial role in various fields of mathematics, physics, engineering, and economics. Understanding the derivatives of logarithmic functions is essential for solving problems involving growth and decay, optimization, and in the study of functions and their rates of change.

Basic Derivative of Logarithmic Functions

The most basic logarithmic function is the natural logarithm, denoted as $\ln(x)$, which is the inverse of the exponential function $e^x$. The derivative of the natural logarithm is given by:

$$ \frac{d}{dx}[\ln(x)] = \frac{1}{x} $$

for $x > 0$.

Derivative of Logarithmic Functions with Different Bases

Logarithmic functions can have bases other than $e$. The general form of a logarithmic function with base $a$ is $\log_a(x)$. The derivative of a logarithmic function with any base $a$ is given by:

$$ \frac{d}{dx}[\log_a(x)] = \frac{1}{x \ln(a)} $$

where $a > 0$ and $a \neq 1$.

Chain Rule and Logarithmic Differentiation

When dealing with more complex logarithmic functions, such as those involving a function of $x$ inside the logarithm, we use the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

For a function of the form $\ln(g(x))$, the derivative is:

$$ \frac{d}{dx}[\ln(g(x))] = \frac{1}{g(x)} \cdot g'(x) $$

where $g'(x)$ is the derivative of $g(x)$.

Table of Differences and Important Points

Property	Natural Logarithm $\ln(x)$	General Logarithm $\log_a(x)$
Definition	Inverse of $e^x$	Inverse of $a^x$
Domain	$x > 0$	$x > 0$
Base	$e$ (Euler's number)	Any positive number $a \neq 1$
Derivative	$\frac{d}{dx}[\ln(x)] = \frac{1}{x}$	$\frac{d}{dx}[\log_a(x)] = \frac{1}{x \ln(a)}$
Chain Rule	$\frac{d}{dx}[\ln(g(x))] = \frac{1}{g(x)} \cdot g'(x)$	$\frac{d}{dx}[\log_a(g(x))] = \frac{1}{g(x) \ln(a)} \cdot g'(x)$

Examples

Example 1: Basic Derivative

Find the derivative of the function $f(x) = \ln(x)$.

Using the basic derivative formula, we get:

$$ f'(x) = \frac{d}{dx}[\ln(x)] = \frac{1}{x} $$

Example 2: Derivative with Different Base

Find the derivative of the function $f(x) = \log_3(x)$.

Using the derivative formula for logarithms with base $a$, we have:

$$ f'(x) = \frac{d}{dx}[\log_3(x)] = \frac{1}{x \ln(3)} $$

Example 3: Applying the Chain Rule

Find the derivative of the function $f(x) = \ln(x^2 + 1)$.

Applying the chain rule, we get:

$$ f'(x) = \frac{d}{dx}[\ln(x^2 + 1)] = \frac{1}{x^2 + 1} \cdot \frac{d}{dx}[x^2 + 1] = \frac{1}{x^2 + 1} \cdot 2x = \frac{2x}{x^2 + 1} $$

Example 4: Logarithmic Differentiation

Find the derivative of the function $f(x) = \ln(\sqrt{x})$.

First, rewrite the function using properties of logarithms:

$$ f(x) = \ln(x^{1/2}) = \frac{1}{2} \ln(x) $$

Now, differentiate using the basic derivative formula:

$$ f'(x) = \frac{1}{2} \cdot \frac{d}{dx}[\ln(x)] = \frac{1}{2} \cdot \frac{1}{x} = \frac{1}{2x} $$

Understanding the derivatives of logarithmic functions is essential for calculus and its applications. By mastering these concepts, you can solve a wide range of problems involving the rates of change of logarithmic relationships.