Derivatives of Exponential and Logarithmic Functions in Business Calculus

Rules of Differentiation for Exponential and Logarithmic Functions

Introduction

Exponential and logarithmic functions are fundamental in business calculus, especially for modeling growth, decay, and rates of change. Understanding their derivatives is essential for solving real-world problems in economics, finance, and the life sciences.

Rules of Differentiation Table

The following table summarizes the main differentiation rules for exponential and logarithmic functions:

Name	Rule	Example
General Exponential	$\frac{d}{dx} b^{x} = b^{x} \cdot \ln b$ $b > 0,\ b \neq 1$	$\frac{d}{dx}(6^{x})$
Natural Exponential	$\frac{d}{dx} e^{x} = e^{x}$	$\frac{d}{dx}(e^{x})$
General Logarithmic	$\frac{d}{dx} \log_{b} x = \frac{1}{x \ln b}$ $b > 0,\ b \neq 1$	$\frac{d}{dx}(\log_{6} x)$
Natural Logarithmic	$\frac{d}{dx} \ln x = \frac{1}{x}$ $x > 0$	$\frac{d}{dx}(6 \ln x)$

Derivatives of General Exponential Functions

Definition and Derivation

The derivative of a general exponential function $b^{x}$ (where $b > 0$, $b \neq 1$) can be found using limits:

• Limit Definition: $\frac{d}{dx}(b^{x}) = \lim_{h \to 0} \frac{b^{x+h} - b^{x}}{h}$

• By properties of exponents: $= \lim_{h \to 0} \frac{b^{x} (b^{h} - 1)}{h}$

• Factoring out $b^{x}$: $= b^{x} \cdot \lim_{h \to 0} \frac{b^{h} - 1}{h}$

• The limit $\lim_{h \to 0} \frac{b^{h} - 1}{h} = \ln b$

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

Example

• Find the derivative of $f(x) = 6^{x}$:

• $f'(x) = 6^{x} \ln 6$

Application

• Exponential functions model compound interest, population growth, and radioactive decay in business and science.

Derivatives of the Natural Exponential Function ($e^{x}$)

Definition

The natural exponential function $e^{x}$ is a special case of $b^{x}$ where $b = e$ (Euler's number, approximately 2.71828).

• Rule: $\frac{d}{dx}(e^{x}) = e^{x}$

Chain Rule for Exponential Functions

• If $f(x) = e^{g(x)}$, then $f'(x) = e^{g(x)} \cdot g'(x)$

Example

• Find the derivative of $f(x) = 3e^{2x+4}$:

• $f'(x) = 3e^{2x+4} \cdot 2 = 6e^{2x+4}$

Applications: Instantaneous Rate of Change

Example: Radioactive Decay

• Given $R = 150 \left(\frac{1}{3}\right)^{t/7}$, find $\frac{dR}{dt}$.

• Apply the chain rule and exponential differentiation to find the instantaneous rate of change at specific times.

Derivatives of General Logarithmic Functions

Definition and Derivation

• For $y = \log_{b} x$, rewrite as $x = b^{y}$.

• Differentiating both sides with respect to $x$ and solving for $\frac{dy}{dx}$ gives:

• $\frac{d}{dx}(\log_{b} x) = \frac{1}{x \ln b}$

Example

• Find the derivative of $g(x) = \log_{5} x$:

• $g'(x) = \frac{1}{x \ln 5}$

Derivatives of the Natural Logarithmic Function ($\ln x$)

Definition

• The natural logarithm is $\ln x = \log_{e} x$.

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

Chain Rule for Logarithmic Functions

• If $f(x) = \ln(g(x))$, then $f'(x) = \frac{1}{g(x)} \cdot g'(x)$

Example

• Find the derivative of $f(x) = \ln(x^{2} + 4x)$:

• $f'(x) = \frac{1}{x^{2} + 4x} \cdot (2x + 4)$

Logarithmic Differentiation

Introduction

Logarithmic differentiation is a technique used to differentiate complicated functions, especially those involving products, quotients, or variable exponents.

• Take the natural logarithm of both sides: $\ln y = \ln(f(x))$

• Expand using log properties (product, quotient, power rules)

• Differentiate both sides implicitly

• Solve for $\frac{dy}{dx}$

Example

• Find the derivative of $y = \frac{(x+4)(x+2)^{5}}{(x^{2}-1)^{2/3}}$ using logarithmic differentiation:

• $\ln y = \ln(x+4) + 5\ln(x+2) - \frac{2}{3}\ln(x^{2}-1)$

• Differentiating both sides and solving for $y'$ gives the derivative.

Summary Table: Rules of Differentiation

Name	Rule	Example
General Exponential	$\frac{d}{dx} b^{x} = b^{x} \ln b$	$\frac{d}{dx}(6^{x}) = 6^{x} \ln 6$
Natural Exponential	$\frac{d}{dx} e^{x} = e^{x}$	$\frac{d}{dx}(e^{x}) = e^{x}$
General Logarithmic	$\frac{d}{dx} \log_{b} x = \frac{1}{x \ln b}$	$\frac{d}{dx}(\log_{6} x) = \frac{1}{x \ln 6}$
Natural Logarithmic	$\frac{d}{dx} \ln x = \frac{1}{x}$	$\frac{d}{dx}(6 \ln x) = \frac{6}{x}$

Practice Problems

• Find the derivative of $f(x) = 2^{x} - 5^{x}$

• Find the derivative of $g(x) = x^{4} + 4x + 4^{x}$

• Find the derivative of $y = (4x - 3x^{2} + 9) \cdot 2^{x}$

• Find the derivative of $h(x) = e^{(x^{2}+3x)}$

• Find the derivative of $f(x) = -3e^{x} + 5x - 2$

• Find the derivative of $g(x) = 7e^{x} + 2x^{3}$

• Find the derivative of $y = x^{2}e^{3x^{2}+5x}$

• Find the derivative of $f(t) = \frac{e^{t}}{t - 2e^{-t}}$

• Find the derivative of $f(x) = 8\log_{2} x$

• Find the derivative of $g(t) = 2t + \log_{5} t$

• Find the derivative of $f(x) = (x^{2} + 2x) \cdot \log_{5} x$

• Find the derivative of $g(t) = \log_{5}(7^{2t+4})$

• Find the derivative of $f(x) = 2x^{2} - 1 + \ln x$

• Find the derivative of $g(x) = e^{x} + \ln x^{5}$

• Find the derivative of $h(x) = \ln\left(\frac{\sqrt{x+1}}{x^{2}+3}\right)$

• Find the derivative of $y = x^{2} \ln(x^{2})$

• Find the derivative of $g(x) = e^{x} \ln x$

• Use logarithmic differentiation to find the derivative of $y = (\sqrt{x+1})^{x}$

• Use logarithmic differentiation to find the derivative of $y = \sqrt{\frac{(x^{2}+2x)^{5}}{x(x^{2}-4)}}$

Applications in Business Calculus

• Exponential and logarithmic derivatives are used to model compound interest, population growth, depreciation, and other business-related phenomena.

• Instantaneous rate of change provides information about marginal cost, marginal revenue, and other economic rates.

Additional info: The notes include both worked examples and practice problems, as well as summary tables for quick reference. The content is directly relevant to Business Calculus, focusing on differentiation techniques for exponential and logarithmic functions, which are essential for business applications.