Derivatives of Exponential and Logarithmic Functions: Business Calculus Study Guide

Derivatives of Exponential and Logarithmic Functions

The Derivative of $e^x$

The exponential function $e^x$ is fundamental in calculus, especially in business applications involving growth and decay. The process of finding its derivative relies on its unique properties.

• Key Point 1: The derivative of $e^x$ is $e^x$ itself.

• Key Point 2: The power rule does not apply to $e^x$; instead, use the definition of the exponential function.

• Example: If $f(x) = e^x$, then $f'(x) = e^x$.

• Caution: When using calculators or technology, ensure you use the designated $e$ for the exponential function.

Four-Step Derivative Process for $e^x$:

1. Find $f(x + h) = e^{x+h} = e^x \cdot e^h$

2. Find $f(x + h) - f(x) = e^x \cdot e^h - e^x = e^x (e^h - 1)$

3. Divide by $h$: $\frac{e^x (e^h - 1)}{h}$

4. Take the limit as $h \to 0$: $\lim_{h \to 0} \frac{e^x (e^h - 1)}{h} = e^x$

Review of Properties of Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and are widely used in business calculus for modeling growth, decay, and other phenomena.

• Key Point 1: For $b > 0$ and $b \neq 1$, $y = \log_b x$ is equivalent to $x = b^y$.

• Key Point 2: The domain of $y = \log_b x$ is $(0, \infty)$; the range is $(-\infty, \infty)$.

• Key Point 3: The domain of $y = b^x$ is $(-\infty, \infty)$; the range is $(0, \infty)$.

• Example: $\log_{10} x$ is the common logarithm; $\ln x = \log_e x$ is the natural logarithm.

The graphs of $y = \log_b x$ and $y = b^x$ are symmetric with respect to the line $y = x$.

Derivative of $\ln x$

The natural logarithm function $\ln x$ is essential in calculus. Its derivative is derived using properties of logarithms.

• Key Point 1: The derivative of $\ln x$ is $\frac{1}{x}$ for $x > 0$.

• Key Point 2: Use logarithmic properties to simplify expressions before differentiating.

• Example: If $f(x) = \ln x$, then $f'(x) = \frac{1}{x}$.

Four-Step Derivative Process for $\ln x$:

1. Find $f(x + h) = \ln(x + h)$

2. Find $f(x + h) - f(x) = \ln(x + h) - \ln(x) = \ln\left(\frac{x + h}{x}\right)$

3. Divide by $h$: $\frac{\ln\left(\frac{x + h}{x}\right)}{h}$

4. Take the limit as $h \to 0$: $\lim_{h \to 0} \frac{\ln\left(1 + \frac{h}{x}\right)}{h} = \frac{1}{x}$

Examples: Finding Derivatives

• Example (A): $y = 3e^x + 5 \ln x$ Derivative: $y' = 3e^x + \frac{5}{x}$

• Example (B): $y = x^4 - \ln x^4$ Rewrite using logarithm properties: $y = x^4 - 4 \ln x$ Derivative: $y' = 4x^3 - \frac{4}{x}$

Other Logarithmic and Exponential Functions

While the base $e$ is preferred in most applications, other bases may be used. Derivatives of $y = \log_b x$ and $y = b^x$ can be found by expressing them in terms of natural logarithms and exponentials.

• Key Point 1: $\frac{d}{dx} \log_b x = \frac{1}{x \ln b}$

• Key Point 2: $\frac{d}{dx} b^x = b^x \ln b$

Change of Base for Logarithms

To relate logarithms of different bases, use the change of base formula:

• Key Point: $\log_b x = \frac{\ln x}{\ln b}$

• Example: $\log_2 8 = \frac{\ln 8}{\ln 2}$

Change of Base for Exponential Functions

Exponential functions with base $b$ can be rewritten in terms of $e$:

• Key Point: $b^x = e^{x \ln b}$

• Example: $2^x = e^{x \ln 2}$

Summary Table: Derivatives of Exponential and Logarithmic Functions

Function	Derivative
$e^x$	$e^x$
$a^x$	$a^x \ln a$
$\ln x$	$\frac{1}{x}$
$\log_b x$	$\frac{1}{x \ln b}$

Examples: Applications in Business Calculus

Exponential Model: Price Demand Equation

In business, exponential models are used to describe price-demand relationships. For example, if $p = 12.5(0.998)^x$, the rate of change of price with respect to demand can be found using derivatives.

• Key Point: The derivative gives the rate at which price changes as demand changes.

• Example: When $x = 100$, the price per set of strings is decreasing at a rate of about 2 cents per set.

Continuous Compound Interest

Continuous compounding is modeled by exponential functions. The instantaneous rate of change of an investment is found using derivatives.

• Key Point: $A(t) = 10,000e^{0.04t}$; $A'(t) = 400e^{0.04t}$

• Example (A): After 2 years, the account grows at $433.31$ per year.

• Example (B): When the balance is $20,000$, the rate of change is $800$ per year (4% of the balance).

Logarithm Model: Franchise Growth

Logarithmic models are used to describe growth over time, such as the number of franchise locations.

• Key Point: $N(t) = -765 + 482 \ln t$ models the number of locations.

• Example: In 2028 ($t = 48$), $N(48) \approx 1,101$ locations.

• Rate of Change: $N'(t) = \frac{482}{t}$; $N'(48) \approx 10$ locations per year in 2028.