Chapter 3: Additional Derivative Topics

Section 3.3: Derivatives of Products and Quotients

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 various applications.

Derivatives of Products

While the derivative of a sum or difference is simply the sum or difference of the derivatives, the same does not hold for products of functions. The derivative of a product is not the product of the derivatives.

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

• Key Point 2: The correct formula for the derivative of a product is given by the Product Rule.

Product Rule (Theorem 1): If and are differentiable functions, then the derivative of their product is:

• Mnemonic: "First times the derivative of the second, plus the second times the derivative of the first."

Example: If and , then

Alternative Method: Sometimes, multiplying the functions first and then differentiating can be simpler, but the product rule is essential when direct multiplication is not practical.

Applications: Tangent Lines

The product rule is often used to find the slope of the tangent line to a curve at a specific point.

• Key Point: The slope of the tangent line at is given by .

• Example: If , then . At , the slope is .

• Horizontal Tangents: The tangent line is horizontal where .

Derivatives of Quotients

The derivative of a quotient is not the quotient of the derivatives. Instead, it follows a specific rule known as the Quotient Rule.

• Key Point 1: The derivative of a quotient is more complex than simply dividing the derivatives.

• Key Point 2: The Quotient Rule provides the correct formula.

Quotient Rule (Theorem 2): If and are differentiable and , then

• Mnemonic: "Denominator times the derivative of the numerator, minus numerator times the derivative of the denominator, all over the denominator squared."

Example: If and , then

Applications: Business Example – Sales Analysis

Derivatives of products and quotients are used in business to analyze rates of change, such as sales growth over time.

• Example: Suppose total sales (in thousands) of a video game months after release are given by .

• To find the rate of change of sales at :

• At , (thousand games), (thousand games per month).

• Interpretation: After 10 months, total sales are 62,500 games, and sales are increasing at a rate of 6,250 games per month.

• Estimation: To estimate sales after 11 months, use (linear approximation).

Additional info: The product and quotient rules are foundational for more advanced calculus topics, including optimization and marginal analysis in business applications.