Section 3.2 - Product Quotient Rules

3.2 The Product and Quotient Rules

Product Rule

• Purpose: To find the derivative of the product of two functions.

• Definition: If F(x) = f(x) · g(x), then the derivative is:

  • F′(x) = f′(x)g(x) + f(x)g′(x)

• Alternative Notation: Letting u = f(x) and v = g(x), then y = uv leads to:

  • y′ = u′v + uv′

Proof of Product Rule

• Start: [f(x) · g(x)]′ = lim h→0 [f(x + h)g(x + h) − f(x)g(x)]/h

• Add and subtract the term f(x + h)g(x) in the limit:

  • = lim h→0 [f(x + h)g(x + h) − f(x + h)g(x) + f(x + h)g(x) − f(x)g(x)]/h

  • = lim h→0 [f(x + h)(g(x + h)−g(x))/h + g(x) (f(x + h)−f(x))/h]

• As h approaches 0, this becomes:

  • f(x)g′(x) + g(x)f′(x) = f′(x)g(x) + f(x)g′(x)

Example 1

• Given: y = x²(x³ + 2)

• Let: u = x², v = x³ + 2

• Derivatives:

  • u′ = 2x, v′ = 3x²

• Apply Product Rule:

  • y′ = u′v + uv′ = 2x(x³ + 2) + x²(3x²)

• Note: Simplification optional.

Limitations of Product Rule

• Some might prefer to directly expand before differentiation such as with f(x) = x⁵ + 2x².

• The utility of the Product and Quotient rules becomes clearer when dealing with complex functions beyond polynomials.

• Example: h(x) = (3x² + 4)(2x² + 3)

  • (a) FOIL & derive using power rule.

  • (b) Derive using product rule & simplify.

Examples of Product Rule Derivatives

• (a) h(x) = x³(√x − 7)

• (b) h(x) = (x⁴ - 3x³ + 5x²)(3x - 6)

• (c) h(x) = (x⁸ - x⁴)(x⁷ - x⁵ + x³)

• (d) h(x) = (x³/2 − x⁻²)(√5x + 1/x⁴)

Quotient Rule

• Purpose: To find the derivative of the quotient of two functions.

• Definition: If F(x) = f(x)/g(x), then the derivative is:

  • F′(x) = [g(x)f′(x) − f(x)g′(x)] / [g(x)]²

• Alternative Notation: Using u = f(x) and v = g(x):

  • y′ = (vu′ − uv′) / v²

Proof of Quotient Rule

• Start: f(x) / g(x) !′ = lim h→0 [f(x+h)g(x) − f(x)g(x+h)]/h

• Proceed similar way as with the product rule, arriving finally at:

  • g(x)f′(x) − f(x)g′(x) / [g(x)]²

Mnemonic for Quotient Rule

• "Low d-high minus high d-low all over the square of what’s below."

Example 3: Using Quotient Rule

• Given: y = (x² + 1) / (x − 1)

• Let: u = x² + 1, v = x − 1

• Derivatives: u′ = 2x, v′ = 1

• Apply the quotient rule:

  • y′ = [(x − 1)(2x) − (x² + 1)(1)] / (x − 1)²

• Result: y′ expression after simplification.

Further Quotient Rule Examples

• (a) h(x) = 5x² / (3x + 5)

• (b) h(x) = (x³ − x) / (x³ + x)

• (c) h(x) = 100x² / (4x² + 10)

Advanced Applications of Quotient Rule

• The quotient rule's utility with exponential and logarithmic functions:

• Example 4: y = x * eˣ

  • Derivatives of u and v.

• Example 5: F(x) = ln(x) / x²

  • Use derived formula of the quotient rule.