Derivatives of Exponential (Base-e) Functions and Business Applications

Chapter 2: Exponential and Logarithmic Functions

Introduction

This section introduces the differentiation of exponential functions, particularly those with base e, and explores their applications in business contexts such as continuous growth models.

Derivatives of Exponential (Base-e) Functions

Objectives

• To differentiate exponential (base-e) functions.

• To solve applied problems involving exponential (base-e) functions and their derivatives.

Theorem 5: Derivative of $f(x) = e^x$

The derivative of the exponential function $f(x) = e^x$ is the function itself:

• Formula:

$\frac{d}{dx} e^x = e^x$

• Key Point: The exponential function $e^x$ is unique in that its rate of change at any point equals its value at that point.

Example 1: Differentiating Exponential Functions

• a) $y = 3e^x$

  • $\frac{dy}{dx} = 3 \frac{d}{dx} e^x = 3e^x$

• b) $y = x^2 e^x$

  • Use the product rule: $\frac{d}{dx}(x^2 e^x) = x^2 \frac{d}{dx} e^x + e^x \frac{d}{dx} x^2 = x^2 e^x + 2x e^x = e^x(x^2 + 2x)$

• c) $\frac{e^x}{x^3}$

  • Use the quotient rule: $\frac{d}{dx} \left( \frac{e^x}{x^3} \right ) = \frac{x^3 e^x - e^x 3x^2}{x^6} = \frac{e^x(x^3 - 3x^2)}{x^6}$

  • Simplify: $\frac{e^x(x - 3)}{x^4}$

Quick Check 1: Practice Problems

• a) $y = 6e^x$

  • $\frac{dy}{dx} = 6e^x$

• b) $y = x^3 e^x$

  • $\frac{dy}{dx} = 3x^2 e^x + x^3 e^x = e^x(x^3 + 3x^2)$

• c) $\frac{e^x}{x^2}$

  • $\frac{d}{dx} \left( \frac{e^x}{x^2} \right ) = \frac{x^2 e^x - e^x 2x}{x^4} = \frac{e^x(x^2 - 2x)}{x^4}$

  • Simplify: $\frac{e^x(x - 2)}{x^3}$

Theorem 6: Derivative of $e^{f(x)}$ (Chain Rule)

For any differentiable function $f(x)$:

• Formula:

$\frac{d}{dx} e^{f(x)} = e^{f(x)} \cdot f'(x)$

• Or, for $u = u(x)$:

$\frac{d}{dx} e^{u} = e^{u} \frac{du}{dx}$

• Key Point: The chain rule allows differentiation of composite exponential functions.

Example 2: Differentiating Composite Exponential Functions

• a) $y = e^{8x}$

  • $\frac{d}{dx} e^{8x} = e^{8x} \cdot 8 = 8e^{8x}$

• b) $y = e^{-x^2 + 4x - 7}$

  • $\frac{d}{dx} e^{-x^2 + 4x - 7} = e^{-x^2 + 4x - 7} \cdot (-2x + 4)$

• c) $y = e^{\sqrt{x^2 - 3}}$

  • Let $u = \sqrt{x^2 - 3}$, $\frac{du}{dx} = \frac{1}{2}(x^2 - 3)^{-1/2} \cdot 2x = \frac{x}{\sqrt{x^2 - 3}}$

  • $\frac{d}{dx} e^{\sqrt{x^2 - 3}} = e^{\sqrt{x^2 - 3}} \cdot \frac{x}{\sqrt{x^2 - 3}}$

Quick Check 2: Practice Problems

• a) $y = e^{-4x}$

  • $\frac{d}{dx} e^{-4x} = -4e^{-4x}$

• b) $y = e^{x^3 + 8x}$

  • $\frac{d}{dx} e^{x^3 + 8x} = e^{x^3 + 8x}(3x^2 + 8)$

• c) $y = e^{\sqrt{x^2 + 5}}$

  • $\frac{d}{dx} e^{\sqrt{x^2 + 5}} = e^{\sqrt{x^2 + 5}} \cdot \frac{x}{\sqrt{x^2 + 5}}$

Example 3: Second Derivative of an Exponential Function

• Find $\frac{d^2 y}{dx^2}$ for $y = e^{-5x^2}$

  • First derivative: $\frac{dy}{dx} = e^{-5x^2} \cdot (-10x) = -10x e^{-5x^2}$

  • Second derivative: $\frac{d^2 y}{dx^2} = \frac{d}{dx}(-10x e^{-5x^2})$

  • Apply product rule: $= (-10x)(-10x e^{-5x^2}) + (e^{-5x^2})(-10)$

  • Simplify: $= 10e^{-5x^2}(10x^2 - 1)$

Business Application: Growth of an Account

Continuous Compounding Formula

When money is invested at a continuous interest rate, the value of the account after $t$ years is given by:

$A(t) = P e^{rt}$

• P: Principal (initial investment)

• r: Annual interest rate (as a decimal)

• t: Time in years

Example 4: Franco's Fishing Emporium

• Initial investment: $P = $50,000

• Interest rate: $r = 0.0125$ (1.25%)

• Value after 5 years:

$A(5) = 50,000 e^{0.0125 \times 5} \approx 53,224.72$

• Interpretation: After 5 years, the account is worth approximately $53,224.72.

Rate of Growth at $t = 5$

• To find the rate at which the account is growing at $t = 5$, compute the derivative:

$A'(t) = \frac{d}{dt} (50,000 e^{0.0125t}) = 50,000 e^{0.0125t} \cdot 0.0125 = 625 e^{0.0125t}$

$A'(5) = 625 e^{0.0125 \times 5} \approx 665.31$

• Interpretation: At $t = 5$ years, the account is increasing at a rate of $665.31 per year.

Summary Table: Rules for Differentiating Exponential Functions

Function	Derivative	Rule Used
$e^x$	$e^x$	Basic Exponential Rule
$a e^x$	$a e^x$	Constant Multiple Rule
$e^{f(x)}$	$e^{f(x)} f'(x)$	Chain Rule
$f(x) e^x$	$f'(x) e^x + f(x) e^x$	Product Rule
$\frac{e^x}{g(x)}$	$\frac{g(x) e^x - e^x g'(x)}{[g(x)]^2}$	Quotient Rule

Additional info: The notes focus on differentiation of exponential functions, including the use of the product, quotient, and chain rules, and provide a business application relevant to Business Calculus students.