(L12) Derivatives of Exponential and Logarithmic Functions in Business Calculus

Exponential Functions and Their Derivatives

Definition and Properties

Exponential functions are of the form f(x) = a^x, where a is a positive constant. These functions are fundamental in modeling growth and decay in business applications.

• Key Property: The rate of change of an exponential function is proportional to its value.

• Special Base: The number e (Euler's number, approximately 2.71828) is defined as $e = ext{lim}_{n o ext{inf}} igg(1 + rac{1}{n}igg)^n$.

Derivative of Exponential Functions

The derivative of f(x) = a^x can be found using the limit definition and the chain rule.

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

• Special Case: For a = e, $\frac{d}{dx}(e^x) = e^x$

• Chain Rule for Exponential Functions: If f(x) = a^{g(x)}, then $\frac{d}{dx}(a^{g(x)}) = (\ln a) a^{g(x)} g'(x)$

Examples

• Example 1: $\frac{d}{dx}(e^{x^2}) = e^{x^2} \cdot 2x$

• Example 2: $\frac{d}{dt}(5^{\sqrt{t}}) = (\ln 5) 5^{\sqrt{t}} \cdot \frac{1}{2\sqrt{t}}$

Logarithmic Functions and Their Derivatives

Definition and Properties

Logarithmic functions are the inverses of exponential functions. The natural logarithm is denoted ln(x), and the logarithm with base a is loga(x).

• Key Property: $a^{\ln x} = x$ for a = e

• Logarithmic Differentiation: Useful for differentiating complicated products, quotients, or powers.

Derivative of Logarithmic Functions

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

• Logarithm with Base a: $\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}$

• Chain Rule for Logarithmic Functions: If f(x) = \ln(g(x)), then $\frac{d}{dx}(\ln(g(x))) = \frac{g'(x)}{g(x)}$

• Generalization with Absolute Values: $\frac{d}{dx}(\ln|x|) = \frac{1}{x}$ for $x \neq 0$

Examples

• Example 1: $\frac{d}{dx}(\ln(kx)) = \frac{1}{x}$, where k is a constant

• Example 2: $\frac{d}{dx}(\log_3 x^3) = (\ln 3) 3x^2 \cdot \frac{1}{x}$

Inverse Functions and Their Derivatives

General Strategy

If g(x) is the inverse of f(x), and you know f'(x), then:

• $f(g(x)) = x \implies f'(g(x)) \cdot g'(x) = 1 \implies g'(x) = \frac{1}{f'(g(x))}$

Example

• Example: If g(x) = \sqrt{x} is the inverse of f(x) = x^2, then $g'(x) = \frac{1}{2\sqrt{x}}$

Logistic Functions in Business Applications

Definition and Application

Logistic functions model growth that starts slowly, increases rapidly, and then levels off as it approaches a maximum (carrying capacity). Common in population dynamics and sales modeling.

• General Form: $P(t) = \frac{mP_0}{P_0 + (m - P_0)e^{-kmt}}$

• Parameters:

  • P_0: Initial population or sales

  • m: Maximum population (carrying capacity)

  • k: Positive constant

• Long-term Behavior: As $t \to \infty$, $P(t) \to m$

Summary Table: Derivatives of Exponential and Logarithmic Functions

Function	Derivative
$e^x$	$e^x$
$a^x$	$(\ln a) a^x$
$e^{g(x)}$	$e^{g(x)} g'(x)$
$a^{g(x)}$	$(\ln a) a^{g(x)} g'(x)$
$\ln x$	$\frac{1}{x}$
$\ln(g(x))$	$\frac{g'(x)}{g(x)}$
$\log_a x$	$\frac{1}{x \ln a}$
$\log_a(g(x))$	$\frac{g'(x)}{g(x) \ln a}$

Additional info:

• These notes cover the essential calculus rules for exponential and logarithmic functions, including the chain rule and applications to business models such as logistic growth.

• Examples and step-by-step differentiation are provided for both simple and composite functions.

• Generalization to absolute values is included for logarithmic derivatives.