Derivatives of Exponential (Base-e) Functions in Business Calculus

Derivatives of Exponential (Base-e) Functions

Exponential Functions: Definition and Properties

Exponential functions are fundamental in calculus and business applications, modeling growth and decay processes. An exponential function is defined as:

• General Form: $f(x) = a_0 \cdot a^x$, where $a > 0$ and $a \neq 1$.

• Base $a$: The constant $a$ is called the base of the exponential function.

• Behavior: The function is continuous and positive for all real $x$.

Case 1: If $a > 1$, the function is increasing and concave up. As $x$ decreases, $f(x)$ approaches 0, and the x-axis is a horizontal asymptote.

Case 2: If $0 < a < 1$, the function is decreasing and concave up. As $x$ increases, $f(x)$ approaches 0, and the x-axis is a horizontal asymptote.

The Natural Base $e$

The number $e$ is a special constant in mathematics, approximately $2.718281828459$, defined by:

• $e = \lim_{h \to 0} (1 + h)^{1/h}$

• $e$ is called the natural base and is used in natural exponential functions.

Derivative of the Natural Exponential Function

The derivative of the natural exponential function $f(x) = e^x$ is unique because it is equal to the function itself:

• $\frac{d}{dx} e^x = e^x$

• In general, $f'(x) = f(x)$ for $f(x) = e^x$

Derivative of $e$ to a Function

When the exponent is a function of $x$, the derivative is the product of $e$ to that function and the derivative of the exponent:

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

• This is a direct application of the chain rule.

Examples of Differentiation

• Example 1: Differentiate $f(x) = e^{4x}$

  • $f'(x) = 4e^{4x}$

• Example 2: Differentiate $f(x) = e^{x^2 + 3x - 9}$

  • $f'(x) = e^{x^2 + 3x - 9} \cdot (2x + 3)$

• Example 3: Differentiate $f(x) = x e^{5x}$

  • Use the product rule: $f'(x) = e^{5x} + 5x e^{5x}$

• Example 4: Differentiate $y = e^{1+x}$

  • $\frac{dy}{dx} = e^{1+x}$

Applied Example: Business Context

Pelican Fabrics purchases a new video surveillance system. The value of the system $V(t)$, in dollars, $t$ years after purchase is modeled by:

• $V(t) = 17,500 e^{-0.0834 t}$

• Estimating Value After 5 Years: Substitute $t = 5$ into the formula.

• Rate of Change After 5 Years: Compute $V'(t)$ and evaluate at $t = 5$.

Example: $V'(t) = 17,500 \cdot (-0.0834) e^{-0.0834 t}$

At $t = 5$:

• $V'(5) = 17,500 \cdot (-0.0834) e^{-0.0834 \cdot 5}$

Summary Table: Properties of Exponential Functions

Base $a$	Behavior	Asymptote	Concavity
$a > 1$	Increasing	x-axis	Up
$0 < a < 1$	Decreasing	x-axis	Up

Practice Problems

• Textbook/Study Plan Practice Problems: 1–23 odd, 55, 75