Derivatives of Products and Quotients- Chapter 3

Section 3.3 Derivatives of Products and Quotients

Introduction

This section explores the rules for differentiating products and quotients of functions, which are essential tools in business calculus for analyzing rates of change in complex expressions. Unlike sums and differences, the derivative of a product or quotient is not simply the product or quotient of the derivatives.

Derivatives of Products

When differentiating the product of two functions, a special rule called the Product Rule must be used. The derivative of a product is not equal to the product of the derivatives.

• Key Point 1: The derivative of a product of two functions is not the product of their derivatives.

• Key Point 2: The Product Rule states: The derivative of the product of two functions is the first function times the derivative of the second, plus the second function times the derivative of the first.

Theorem 1 (Product Rule):

If $f(x)$ and $g(x)$ are differentiable functions, then:

$\frac{d}{dx}[f(x)g(x)] = f(x) \cdot g'(x) + g(x) \cdot f'(x)$

Example: If $f(x) = x^2$ and $g(x) = \sin(x)$, then:

$\frac{d}{dx}[x^2 \sin(x)] = x^2 \cdot \cos(x) + \sin(x) \cdot 2x$

Application: The Product Rule is used in business calculus to differentiate revenue, cost, or profit functions that are products of two variables.

Example 1: Differentiating a Product

Two methods can be used to differentiate a product:

1. Apply the Product Rule directly.

2. Expand the product first, then differentiate each term.

Example: For $f(x) = (x+1)(x^2)$, Method 1 uses the Product Rule, Method 2 expands to $x^3 + x^2$ and differentiates term by term.

Example 2: Tangent Lines

Finding the equation of a tangent line involves differentiating the function and evaluating the derivative at a specific point.

• Key Point: The slope of the tangent line at $x = a$ is $f'(a)$.

• Point-Slope Form: The equation is $y - f(a) = f'(a)(x - a)$.

Example: If $f(x) = x^3 - 3x^2 + 4x$, the slope at $x = 3$ is $f'(3) = 12$.

Horizontal Tangents: The tangent line is horizontal where $f'(x) = 0$.

Example: Solve $f'(x) = 0$ to find where the tangent is horizontal.

Derivatives of Quotients

When differentiating the quotient of two functions, the Quotient Rule must be used. The derivative of a quotient is not equal to the quotient of the derivatives.

• Key Point 1: The derivative of a quotient of two functions is not the quotient of their derivatives.

• Key Point 2: The Quotient Rule states: The derivative of a quotient is the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, divided by the denominator squared.

Theorem 2 (Quotient Rule):

If $f(x)$ and $g(x)$ are differentiable functions, then:

$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{g(x) \cdot f'(x) - f(x) \cdot g'(x)}{[g(x)]^2}$

Example: If $f(x) = x^2$ and $g(x) = x+1$, then:

$\frac{d}{dx}\left[\frac{x^2}{x+1}\right] = \frac{(x+1) \cdot 2x - x^2 \cdot 1}{(x+1)^2}$

Application: The Quotient Rule is used in business calculus to differentiate ratios such as average cost or profit per unit.

Example 3 & 4: Differentiating Quotients

Apply the Quotient Rule to find derivatives of functions expressed as quotients.

Example 5: Sales Analysis

Business calculus often applies derivative rules to real-world problems such as sales analysis. If total sales $S$ (in thousands of games) as a function of time $t$ (months) is given, the derivative $S'(t)$ represents the rate of change of sales.

• Key Point: $S'(t)$ gives the rate at which sales are increasing or decreasing at time $t$.

• Application: Use $S'(t)$ to estimate future sales and interpret business performance.

Example: If $S(10) = 62,500$ games and $S'(10) = 6,250$ games/month, then after 11 months, estimated sales are $S(10) + S'(10) = 68,750$ games.

Additional info: The included image is the cover of the textbook, which is directly relevant as it identifies the source and context for the business calculus material.