Techniques of Differentiation: Product and Quotient Rules

Objectives and Overview

This section covers essential techniques for differentiating functions, focusing on the product and quotient rules. Students will learn to apply these rules, combine them with other differentiation techniques, and solve applied problems involving rates of change and tangent lines.

• Apply the power rule for derivatives.

• Apply sum and constant multiple rules for derivatives.

• Apply exponential rules (any base) for derivatives.

• Calculate derivatives using a combination of rules.

• Use derivatives to solve questions involving:

  • Higher order derivatives

  • Treating certain variables as constant

  • Rates of change with correct units

  • Position, velocity, and acceleration with correct units

  • Tangent lines

Product Rule for Differentiation

Definition and Formula

The product rule is used to differentiate the product of two functions. If f(x) and g(x) are differentiable, then:

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

Applying the Product Rule

• Identify the two functions being multiplied.

• Differentiate each function separately.

• Apply the product rule formula.

Example: Find the derivative of $h(x) = x e^{x}$.

Let $f(x) = x$ and $g(x) = e^{x}$.

$h'(x) = x \cdot \frac{d}{dx}(e^{x}) + e^{x} \cdot \frac{d}{dx}(x) = x e^{x} + e^{x}$

Practice Problems

• Differentiate $x \sin x$

• Differentiate $e^{x} \cos x$

• Differentiate $x^{2} e^{x}$

Quotient Rule for Differentiation

Definition and Formula

The quotient rule is used to differentiate the quotient of two functions. If $H(x)$ and $L(x)$ are differentiable, then:

$\frac{d}{dx}\left[\frac{H(x)}{L(x)}\right] = \frac{L(x)H'(x) - H(x)L'(x)}{[L(x)]^2}$

Note: The product and quotient rules work for multiplication and division of differentiable functions.

Applying the Quotient Rule

• Identify the numerator $H(x)$ and denominator $L(x)$.

• Differentiate both $H(x)$ and $L(x)$.

• Apply the quotient rule formula.

Example: Differentiate $y = \frac{3x - 5}{1 + 2x^2}$.

Practice Problem

Differentiate $y = \frac{x^2 e^x}{1 + 3x^4}$.

Applications of the Product and Quotient Rules

Tangent Lines and Rates of Change

Derivatives can be used to find the equation of the tangent line to a curve at a given point and to compute rates of change in applied contexts.

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

• The equation of the tangent line at $(a, f(a))$ is $y = f'(a)(x - a) + f(a)$.

Example: Find the tangent line to $g(x) = \frac{2x}{x^2 + 1}$ at $x = 2$.

Applied Example: Revenue Rate of Change

Suppose a business's revenue $R(t)$ is the product of the number of sales $N(t)$ and the daily price $P(t)$. The rate of change of revenue is found using the product rule:

$R'(t) = N'(t)P(t) + N(t)P'(t)$

Example: If $N(5) = 100$, $N'(5) = 3$, $P(5) = 18$, $P'(5) = -1$, then:

$R'(5) = (3)(18) + (100)(-1) = 54 - 100 = -46$

Additional Practice and Examples

• Differentiate $g(x) = \frac{x^3 - 2}{x}$ using the quotient rule.

Summary Table: Product and Quotient Rules

Rule	Formula	When to Use
Product Rule	$\frac{d}{dx}[f(x)g(x)] = f(x)g'(x) + g(x)f'(x)$	When differentiating a product of two functions
Quotient Rule	$\frac{d}{dx}\left[\frac{H(x)}{L(x)}\right] = \frac{L(x)H'(x) - H(x)L'(x)}{[L(x)]^2}$	When differentiating a quotient of two functions

Additional info: The notes also reference using derivatives for higher order derivatives, treating variables as constants, and interpreting rates of change in applied contexts such as velocity and acceleration. These are standard applications in calculus and are foundational for further study in mathematics, physics, and engineering.