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

Derivatives of Exponential, Logarithmic, and Trigonometric Functions

Definition and Key Properties

Derivatives measure the instantaneous rate of change of a function with respect to its variable. In business calculus, understanding how to differentiate exponential, logarithmic, and trigonometric functions is essential for modeling growth, decay, and periodic phenomena.

• Exponential Functions: The derivative of $f(x) = e^{x}$ is $f'(x) = e^{x}$.

• Logarithmic Functions: The derivative of $f(x) = ext{ln}(x)$ is $f'(x) = \frac{1}{x}$.

• Trigonometric Functions: The derivative of $f(x) = \sin(x)$ is $f'(x) = \cos(x)$; for $f(x) = \cos(x)$, $f'(x) = -\sin(x)$.

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

• $f'(x) = 0 + 2x e^{x^2}$ (using the chain rule).

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

Analyzing Function Behavior

The first derivative provides information about where a function is increasing or decreasing and helps identify critical points (where $f'(x) = 0$ or is undefined).

• 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; potential locations for local maxima or minima.

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

• Compute $f'(x) = 3x^2 - 6x + 2$ and solve $f'(x) = 0$ for critical points.

Second Derivative: Concavity and Inflection Points

Understanding the Shape of Graphs

The second derivative, $f''(x)$, describes the concavity of a function and helps locate inflection points, where the function changes from concave up to concave down or vice versa.

• Concave Up: $f''(x) > 0$

• Concave Down: $f''(x) < 0$

• Inflection Point: A point where $f''(x)$ changes sign.

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

• $f''(x) = 12x^2 - 24x + 12$

Derivative Rules: Product, Quotient, and Chain Rule

Techniques for Differentiation

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

• Product Rule: If $f(x) = u(x)v(x)$, then $f'(x) = u'(x)v(x) + u(x)v'(x)$

• Quotient Rule: If $f(x) = \frac{u(x)}{v(x)}$, then $f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$

• Chain Rule: If $f(x) = g(h(x))$, then $f'(x) = g'(h(x)) \cdot h'(x)$

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

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

Sketching Graphs and Interpreting Derivatives

Connecting Calculus to Graphical Analysis

Graphing functions and their derivatives helps visualize increasing/decreasing behavior, concavity, and locations of extrema and inflection points.

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

• Intervals of Increase/Decrease: Use sign charts for $f'(x)$.

• Concavity: Use sign charts for $f''(x)$.

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

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

Applications of Derivatives in Business Contexts

Modeling and Interpreting Rates of Change

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

• Rate of Change: The derivative represents how a quantity changes with respect to time or another variable.

• Interpretation: In business, this could mean profit growth, cost reduction, or inventory changes.

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 at which the tree is growing at time $t$.

Summary Table: Derivative Rules

Rule	Formula	Example
Product Rule	$f'(x) = u'(x)v(x) + u(x)v'(x)$	$f(x) = x^2 \sin(x) \Rightarrow f'(x) = 2x \sin(x) + x^2 \cos(x)$
Quotient Rule	$f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$	$f(x) = \frac{x^2}{x+1} \Rightarrow f'(x) = \frac{2x(x+1) - x^2}{(x+1)^2}$
Chain Rule	$f'(x) = g'(h(x)) \cdot h'(x)$	$f(x) = e^{x^2} \Rightarrow f'(x) = 2x e^{x^2}$

Practice Problems Overview

Types of Questions Included

• Find the first and second derivatives of various functions (polynomial, exponential, logarithmic, trigonometric).

• Apply derivative rules to composite and rational functions.

• Interpret the meaning of derivatives in applied contexts (e.g., rate of change in business scenarios).

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

• Sketch graphs based on given derivative properties.

Additional info:

• Some problems require interpreting the derivative in context, such as finding the rate of change of a solution's pH level or the growth rate of a tree.

• Graph analysis questions involve estimating intervals and identifying extrema and inflection points from a provided graph.

• Sketching problems require constructing a function with specified derivative properties.