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 applications.

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

• Special Base: The number e ≈ 2.71828 is the unique base for which the derivative of e^x is itself.

Formula:

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

General Exponential Derivative:

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

• ln a is the natural logarithm of the base a.

• This formula is derived using the definition of the derivative and properties of logarithms.

Chain Rule for Exponential Functions

When the exponent is a function of x, the chain rule is used:

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

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

• Here, g(x) is a differentiable function.

• These formulas are essential for modeling compound growth and other business phenomena.

Examples

• Example 1: $\frac{d}{dx}(e^{5x^2}) = e^{5x^2} \cdot 10x$

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

Logistic Functions in Business Applications

Definition and Application

Logistic functions model growth that starts exponentially but levels off as it approaches a maximum value, called the carrying capacity. This is common in population growth and sales saturation.

• General Form:

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

• P(t): Population or sales at time t

• P_0: Initial population or sales

• m: Maximum population (carrying capacity)

• k: Positive constant (growth rate)

As t → ∞, P(t) → m.

Logarithmic Functions and Their Derivatives

Definition and Properties

Logarithmic functions are the inverses of exponential functions. The natural logarithm ln x is the inverse of e^x.

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

Derivative of the Natural Logarithm

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

• Valid for x > 0.

Derivative of Logarithms with Other Bases

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

• log_a x is the logarithm base a.

Chain Rule for Logarithmic Functions

For composite functions:

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

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

• These formulas are used when the argument of the logarithm is a function of x.

Generalization with Absolute Values

For all x ≠ 0:

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

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

Logarithmic Differentiation

Technique and Application

Logarithmic differentiation is useful for finding derivatives of complicated functions, especially those involving products, quotients, or variable exponents.

• Steps:

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

  2. Differentiate both sides with respect to x using implicit differentiation.

  3. Solve for $\frac{dy}{dx}$.

• Example: For $y = x^x$, take $\ln y = x \ln x$, then differentiate:

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

  • $\frac{dy}{dx} = x^x (\ln x + 1)$

Derivatives of Inverse Functions

General Strategy

If g(x) is the inverse of f(x), then:

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

• This is useful for finding derivatives of inverse functions when the derivative of the original function is known.

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

Summary Table: Derivatives of Exponential and Logarithmic Functions

Additional info:

• These notes cover the essential calculus concepts for exponential and logarithmic functions, which are highly relevant for business calculus, especially in modeling growth, decay, and saturation phenomena.

• Logarithmic differentiation is particularly useful for functions with variable exponents or products/quotients of several functions.

• Understanding the chain rule and inverse function derivatives is crucial for more advanced business calculus topics.