Derivatives of Logarithmic and Exponential Functions: 3.9

Derivatives of Logarithmic and Exponential Functions

Introduction

This section explores the differentiation rules for logarithmic and exponential functions, focusing on natural logarithms, logarithms with arbitrary bases, and exponential functions. These rules are essential for solving a wide range of calculus problems involving growth, decay, and complex function compositions.

Natural Logarithm Rules

Properties of the Natural Logarithm

• Product Rule: $\ln(ab) = \ln a + \ln b$

• Quotient Rule: $\ln\left(\frac{a}{b}\right) = \ln a - \ln b$

• Power Rule: $\ln(a^b) = b \ln a$

• Change of Base Formula: $\log_b a = \frac{\ln a}{\ln b}$

These properties allow us to simplify logarithmic expressions before differentiating.

Derivatives of Logarithmic Functions

Derivative of the Natural Logarithm

• Basic Rule: $\frac{d}{dx}(\ln x) = \frac{1}{x}$

• Chain Rule for Logarithmic Derivative: $\frac{d}{dx}(\ln u) = \frac{1}{u} \cdot \frac{du}{dx}$, where $u$ is a differentiable function of $x$.

Examples

• Example 1: $y = \ln(4x)$ $y' = \frac{1}{4x} \cdot 4 = \frac{1}{x}$

• Example 2: $y = \ln|\sec x|$ $y' = \frac{1}{\sec x} \cdot (\sec x \tan x) = \tan x$

• Example 3: $y = \ln(\ln(10x))$ $y' = \frac{1}{\ln(10x)} \cdot \frac{1}{x} = \frac{1}{x \ln(10x)}$

• Example 4: $y = x \ln(2x)$ $y' = x' \cdot \ln(2x) + x \cdot (\ln(2x))' = 1 \cdot \ln(2x) + x \cdot \frac{1}{x} = \ln(2x) + 1$

• Example 5: $y = \ln((x+9)^2 (3x+2)^3)$

  • First, expand using logarithm properties: $y = 2\ln(x+9) + 3\ln(3x+2)$

  • Differentiate: $y' = 2 \cdot \frac{1}{x+9} + 3 \cdot \frac{1}{3x+2} \cdot 3 = \frac{2}{x+9} + \frac{9}{3x+2}$

• Example 6: $y = \ln\left(\frac{3x+1}{2x-5}\right)$

  • Rewrite: $y = \ln(3x+1) - \ln(2x-5)$

  • Differentiate: $y' = \frac{1}{3x+1} \cdot 3 - \frac{1}{2x-5} \cdot 2 = \frac{3}{3x+1} - \frac{2}{2x-5}$

Derivatives of Logarithms with Arbitrary Bases

• General Rule: $\frac{d}{dx}(\log_b x) = \frac{1}{x \ln b}$

• Example 7: $y = \log_3 x$ $y' = \frac{1}{x \ln 3}$

Derivatives of Exponential Functions

• General Rule: $\frac{d}{dx}(a^x) = a^x \ln a$

• Example 8: $y = 3^x$

  • Take natural log: $\ln y = x \ln 3$

  • Differentiate both sides: $\frac{1}{y} y' = \ln 3$

  • So, $y' = 3^x \ln 3$

Logarithmic Differentiation

Logarithmic differentiation is useful for differentiating functions of the form $y = f(x)^{g(x)}$ or products/quotients of several functions.

• Example 9: $y = x^{\tan x}$

  • Take natural log: $\ln y = \tan x \cdot \ln x$

  • Differentiate both sides: $\frac{1}{y} y' = \sec^2 x \ln x + \tan x \cdot \frac{1}{x}$

  • So, $y' = x^{\tan x} \left( \sec^2 x \ln x + \frac{\tan x}{x} \right)$

Summary Table: Logarithmic and Exponential Derivative Rules

Function	Derivative
$\ln x$	$\frac{1}{x}$
$\ln u$	$\frac{1}{u} \cdot \frac{du}{dx}$
$\log_b x$	$\frac{1}{x \ln b}$
$a^x$	$a^x \ln a$
$u^v$ (logarithmic differentiation)	$u^v \left( \frac{v'}{v} \ln u + v \frac{u'}{u} \right)$ Additional info: This is a generalization using the product and chain rules.