Derivative of ln(x): Proof and Examples

This page presents the derivative of the natural logarithm function ln(x) using the definition of the derivative. We also cover the derivative of composite functions of the form ln(u(x)) and provide detailed examples.

Proof of the Derivative of ln(x) Using the Definition

The derivative of a function f is defined as: \[ f'(x) = \lim_{h \to 0} \dfrac{f(x+h)-f(x)}{h} \] Let \( f(x) = \ln(x) \). Then \[ f'(x) = \lim_{h \to 0} \dfrac{\ln(x+h) - \ln(x)}{h} = \lim_{h \to 0} \dfrac{\ln\left(\dfrac{x+h}{x}\right)}{h} \] Using the power property of logarithms, we write \[ f'(x) = \lim_{h \to 0} \ln\left(1+\dfrac{h}{x}\right)^{\dfrac{1}{h}} \] Substitute \( y = \dfrac{h}{x} \), so \( h = y x \) and \( \lim_{h \to 0} y = 0 \). Then \[ f'(x) = \lim_{y \to 0} \dfrac{1}{x} \ln\left(1+y\right)^{\dfrac{1}{y}} = \dfrac{1}{x} \ln(e) = \dfrac{1}{x} \] \[ \boxed{\large{ \color{red}{\dfrac{d}{dx} \ln(x) = \dfrac{1}{x}}} } \]

Derivative of a Composite Function \( y = \ln(u(x)) \)

Using the chain rule: \[ \boxed{\large{ \color{red}{ \dfrac{d}{dx} \ln(u(x)) = \dfrac{1}{u} \dfrac{du}{dx} }}} \]

Example 1: Derivatives of Composite ln Functions

Find the derivatives of the following functions:

  1. \( f(x) = \ln\left(\dfrac{x^2}{x-2}\right) \)
  2. \( g(x) = \ln\left(\sqrt{x^3+1}\right) \)
  3. \( h(x) = \ln(x^2 + 2x - 5) \)

Solutions

  1. Let \( u(x) = \dfrac{x^2}{x-2} \), then \( u'(x) = \dfrac{x^2-4x}{(x-2)^2} \). \[ f'(x) = \dfrac{1}{u} u' = \dfrac{1}{\dfrac{x^2}{x-2}} \cdot \dfrac{x^2-4x}{(x-2)^2} = \dfrac{x-4}{x(x-2)} \]
  2. Let \( u(x) = \sqrt{x^3+1} \), then \( u'(x) = \dfrac{3x^2}{2\sqrt{x^3+1}} \). \[ g'(x) = \dfrac{1}{u} u' = \dfrac{1}{\sqrt{x^3+1}} \cdot \dfrac{3x^2}{2\sqrt{x^3+1}} = \dfrac{3x^2}{2(x^3+1)} \]
  3. Let \( u(x) = x^2+2x-5 \), then \( u'(x) = 2x+2 \). \[ h'(x) = \dfrac{1}{u} u' = \dfrac{2x+2}{x^2+2x-5} \]

References and Further Reading