Derivatives of Exponential and Logarithmic Functions in Business Calculus

Exponential Functions and Their Derivatives

Definition and Properties of Exponential Functions

Exponential functions are of the form f(x) = a^x, where a > 0 is a constant. These functions are fundamental in modeling growth and decay in business, economics, and the natural sciences.

• Key Property: The base a is a positive real number, and the function grows (if a > 1) or decays (if 0 < a < 1) as x increases.

• Special Case: The natural exponential function f(x) = e^x, where e ≈ 2.71828, is especially important due to its unique calculus properties.

Derivative of the Exponential Function

The derivative of an exponential function describes the rate of change of the function with respect to its variable. For f(x) = a^x:

• The derivative at x = 0 is found using the limit definition:

$ f'(0) = \lim_{h \to 0} \frac{a^h - 1}{h} $

• For the natural exponential function f(x) = e^x:

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

• For a general base a:

$ \frac{d}{dx}(a^x) = (\ln a) \cdot a^x $

• Interpretation: The slope of the tangent line to the graph of f(x) = a^x at any point is proportional to the function's value at that point.

Derivative of Exponential Functions with Variable Exponents

When the exponent is a function of x, i.e., f(x) = a^{g(x)}, the chain rule applies:

$ \frac{d}{dx}(a^{g(x)}) = (\ln a) \cdot a^{g(x)} \cdot g'(x) $

• For f(x) = e^{g(x)}:

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

Applications: Logistic Functions

Logistic functions model population growth or sales that start slowly, increase rapidly, and then level off at a maximum value (carrying capacity).

The general form is:

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

• P(t): Population (or sales) at time t

• P_0: Initial population (or sales)

• m: Maximum possible population (carrying capacity)

• k: Positive constant

As t → ∞, P(t) → m.

Logarithmic Functions and Their Derivatives

Definition and Properties of Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. The natural logarithm is f(x) = \ln x, and the logarithm with base a is f(x) = \log_a x.

• Domain: x > 0

• Key Property: e^{\ln x} = x and \ln(e^x) = x

Derivative of the Natural Logarithm

The derivative of the natural logarithm is:

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

• This formula holds for all x > 0.

Derivative of the Logarithm with Arbitrary Base

For f(x) = \log_a x:

$ \frac{d}{dx}(\log_a x) = \frac{1}{x \ln a} $

• Here, \ln a is the natural logarithm of the base.

Derivative of Logarithmic Functions with Variable Arguments

For f(x) = \ln(g(x)):

$ \frac{d}{dx}(\ln(g(x))) = \frac{g'(x)}{g(x)} $

For f(x) = \log_a(g(x)):

$ \frac{d}{dx}(\log_a(g(x))) = \frac{1}{\ln a} \cdot \frac{g'(x)}{g(x)} $

Generalization with Absolute Values

For all x ≠ 0:

$ \frac{d}{dx}(\ln|x|) = \frac{1}{x} $

For f(x) = \log_a|x|:

$ \frac{d}{dx}(\log_a|x|) = \frac{1}{x \ln a} $

For f(x) = \ln|g(x)|:

$ \frac{d}{dx}(\ln|g(x)|) = \frac{g'(x)}{g(x)} $

For f(x) = \log_a|g(x)|:

$ \frac{d}{dx}(\log_a|g(x)|) = \frac{1}{\ln a} \cdot \frac{g'(x)}{g(x)} $

Logarithmic Differentiation

Technique and Application

Logarithmic differentiation is useful for differentiating complicated products, quotients, or powers. The steps are:

1. Take the natural logarithm of both sides: y = f(x) becomes \ln y = \ln f(x).

2. Differentiate both sides with respect to x, using the chain rule on the left.

3. Solve for y' in terms of x.

Example: For y = x^x:

$ \ln y = x \ln x $

$ \frac{1}{y} \frac{dy}{dx} = \ln x + 1 $

$ \frac{dy}{dx} = x^x (\ln x + 1) $

Derivatives of Inverse Functions

General Rule

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: To find the derivative of g(x) = \sqrt{x} (the inverse of f(x) = x^2):

$ f(x) = x^2 \implies f'(x) = 2x $

$ g'(x) = \frac{1}{2\sqrt{x}} $

Summary Table: Derivatives of Exponential and Logarithmic Functions

Function	Derivative	Domain
$e^x$	$e^x$	All real x
$a^x$	$(\ln a) a^x$	All real x, $a > 0$
$e^{g(x)}$	$e^{g(x)} g'(x)$	g(x) defined
$a^{g(x)}$	$(\ln a) a^{g(x)} g'(x)$	g(x) defined, $a > 0$
$\ln x$	$\frac{1}{x}$	$x > 0$
$\log_a x$	$\frac{1}{x \ln a}$	$x > 0$, $a > 0$, $a \neq 1$
$\ln(g(x))$	$\frac{g'(x)}{g(x)}$	$g(x) > 0$
$\log_a(g(x))$	$\frac{1}{\ln a} \cdot \frac{g'(x)}{g(x)}$	$g(x) > 0$, $a > 0$, $a \neq 1$

Examples and Applications

• Example 1: $f(x) = e^{x^2}$ $f'(x) = e^{x^2} \cdot 2x$

• Example 2: $y = 5^{1/t}$ $\frac{d}{dt} y = (\ln 5) \cdot 5^{1/t} \cdot \left(-\frac{1}{t^2}\right)$

• Example 3: $f(x) = \ln(x^2 + x)$ $f'(x) = \frac{2x + 1}{x^2 + x}$

• Example 4: $f(x) = 3^x \cdot \log_3 x$ $f'(x) = (\ln 3) 3^x \log_3 x + 3^x \cdot \frac{1}{x \ln 3}$

Key Points to Remember

• Exponential and logarithmic functions are inverses of each other.

• The chain rule is essential for differentiating compositions involving exponentials and logarithms.

• Logarithmic differentiation is a powerful tool for complex products and powers.

• Always check the domain of the function before applying derivative formulas, especially for logarithms.

• For absolute values, use $\ln|x|$ and adjust the domain accordingly.