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 \( h = yx \) and \( \lim_{h \to 0} y = 0 \). Then:
\[ f'(x) = \lim_{y \to 0} \dfrac{1}{x} \ln\left(1+y\right)^{\dfrac{1}{y}} = \dfrac{1}{x} \ln(e) = \dfrac{1}{x} \]
Formula:
\[ \dfrac{d}{dx} \ln(x) = \dfrac{1}{x} \]
Derivative of a Composite Function \( y = \ln(u(x)) \)
Using the chain rule:
Chain Rule Formula:
\[ \dfrac{d}{dx} \ln(u(x)) = \dfrac{1}{u} \dfrac{du}{dx} \]
Examples with Solutions
Example 1: Derivatives of Composite ln Functions
Find the derivatives of the following functions:
- \( f(x) = \ln\left(\dfrac{x^2}{x-2}\right) \)
- \( g(x) = \ln\left(\sqrt{x^3+1}\right) \)
- \( h(x) = \ln(x^2 + 2x - 5) \)
Solutions:
- Let \( u(x) = \dfrac{x^2}{x-2} \), then \( u'(x) = \dfrac{2x(x-2) - x^2(1)}{(x-2)^2} = \dfrac{x^2-4x}{(x-2)^2} \). \[ f'(x) = \dfrac{1}{u} u' = \dfrac{1}{\dfrac{x^2}{x-2}} \cdot \dfrac{x^2-4x}{(x-2)^2} = \dfrac{x-4}{x(x-2)} \]
- Let \( u(x) = \sqrt{x^3+1} \), then \( u'(x) = \dfrac{3x^2}{2\sqrt{x^3+1}} \). \[ g'(x) = \dfrac{1}{u} u' = \dfrac{1}{\sqrt{x^3+1}} \cdot \dfrac{3x^2}{2\sqrt{x^3+1}} = \dfrac{3x^2}{2(x^3+1)} \]
- Let \( u(x) = x^2+2x-5 \), then \( u'(x) = 2x+2 \). \[ h'(x) = \dfrac{1}{u} u' = \dfrac{2x+2}{x^2+2x-5} \]