Business Calculus: Derivatives, Applications, and Graphical Analysis Study Guide

Derivatives of Exponential, Logarithmic, and Trigonometric Functions

Definition and Key Properties

Derivatives measure the instantaneous rate of change of a function. In business calculus, exponential, logarithmic, and trigonometric functions frequently model growth, decay, and periodic phenomena.

• Exponential Functions: Functions of the form $f(x) = a e^{bx}$, where $a$ and $b$ are constants.

• Logarithmic Functions: Functions of the form $f(x) = ext{ln}(x)$ or $f(x) = ext{log}_b(x)$.

• Trigonometric Functions: Functions such as $f(x) = ext{sin}(x)$, $f(x) = ext{cos}(x)$, $f(x) = ext{tan}(x)$.

Key Derivative Formulas:

• $\frac{d}{dx} e^{x} = e^{x}$

• $\frac{d}{dx} a^{x} = a^{x} \ln(a)$

• $\frac{d}{dx} \ln(x) = \frac{1}{x}$

• $\frac{d}{dx} \sin(x) = \cos(x)$

• $\frac{d}{dx} \cos(x) = -\sin(x)$

• $\frac{d}{dx} \tan(x) = \sec^2(x)$

Example: Find the derivative of $f(x) = 5 + e^{x^2}$.

• $f'(x) = 0 + \frac{d}{dx} e^{x^2} = e^{x^2} \cdot 2x$ (by chain rule)

Using the First Derivative: Increasing, Decreasing, and Critical Points

Analyzing Function Behavior

The first derivative $f'(x)$ provides information about where a function is increasing or decreasing, and helps identify critical points.

• Increasing: $f'(x) > 0$ on an interval.

• Decreasing: $f'(x) < 0$ on an interval.

• Critical Points: Points where $f'(x) = 0$ or $f'(x)$ is undefined.

Example: For $f(x) = x^3 - 3x^2 + 2x$, find where the function is increasing and decreasing.

• Compute $f'(x) = 3x^2 - 6x + 2$.

• Solve $f'(x) = 0$ for critical points, then test intervals.

Second Derivative: Inflection Points and Concavity

Understanding Concavity and Inflection

The second derivative $f''(x)$ reveals the concavity of a function and helps locate inflection points.

• Concave Up: $f''(x) > 0$

• Concave Down: $f''(x) < 0$

• Inflection Point: Where $f''(x)$ changes sign

Example: For $f(x) = x^4 - 4x^3 + 6x^2 - 2$, find $f''(x)$ and determine inflection points.

• $f'(x) = 4x^3 - 12x^2 + 12x$

• $f''(x) = 12x^2 - 24x + 12$

• Solve $f''(x) = 0$ for possible inflection points.

Derivative Rules: Product, Quotient, and Chain Rule

Applying Derivative Techniques

Complex functions often require the use of derivative rules to compute their derivatives.

• Product Rule: $\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$

• Quotient Rule: $\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right] = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$

• Chain Rule: $\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$

Example: Find the derivative of $h(x) = \ln(1 - 5x^2 + x^3)$.

• Let $u(x) = 1 - 5x^2 + x^3$

• $h'(x) = \frac{1}{u(x)} \cdot u'(x) = \frac{1}{1 - 5x^2 + x^3} \cdot (-10x + 3x^2)$

Graphical Analysis: Sketching and Interpreting Graphs

Connecting Derivatives to Graphs

Graphical analysis involves using derivatives to sketch and interpret the behavior of functions.

• Intervals of Increase/Decrease: Use $f'(x)$ to identify where the graph rises or falls.

• Relative Extrema: Points where $f'(x) = 0$ and the sign of $f'(x)$ changes.

• Concavity and Inflection Points: Use $f''(x)$ to determine where the graph bends upward or downward.

Example: Given a graph of $g(x)$, estimate intervals of increase/decrease and concavity.

Applications of the Derivative in Business Contexts

Modeling and Rate of Change

Derivatives are used to model rates of change in business scenarios, such as growth, decay, and optimization.

• Rate of Change: The derivative $f'(x)$ represents the rate at which a quantity changes with respect to time or another variable.

• Optimization: Finding maximum or minimum values to optimize profit, cost, or other business metrics.

Example: The height $H(t)$ of a tree is modeled by $H(t) = 145(1 - e^{-0.03t})$.

• Find $H'(t)$ to determine the rate of growth at a specific time.

Summary Table: Derivative Rules

Rule	Formula	Example
Product Rule	$\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$	$\frac{d}{dx}[x^2 \sin(x)] = 2x \sin(x) + x^2 \cos(x)$
Quotient Rule	$\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right] = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$	$\frac{d}{dx}\left[\frac{\ln(x)}{x^2}\right] = \frac{\frac{1}{x} \cdot x^2 - \ln(x) \cdot 2x}{x^4}$
Chain Rule	$\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$	$\frac{d}{dx} e^{x^2} = e^{x^2} \cdot 2x$

Practice Problems Overview

Types of Questions

• Find first and second derivatives of various functions.

• Apply derivative rules to exponential, logarithmic, and trigonometric functions.

• Analyze graphs to determine intervals of increase/decrease, concavity, and inflection points.

• Solve business application problems involving rates of change and optimization.

• Sketch graphs based on derivative information.

Additional info:

• Some problems require interpreting the meaning of the derivative in context (e.g., rate of change of a physical or business quantity).

• Graphical analysis is emphasized, including identifying extrema and inflection points from graphs.

• Review includes both computational and conceptual questions, suitable for exam preparation.