Overview
This lecture introduces the product and quotient rules for derivatives, detailing how to differentiate products and quotients of functions, and provides worked examples to illustrate proper application.
The Product Rule
- The product rule states: (fg)' = f'g + fg'.
- To differentiate y = x²eˣ, use the product rule: y' = 2x·eˣ + x²·eˣ.
- For y = x⁴eˣ, the derivative is 4x³eˣ + x⁴eˣ, which factors as x³eˣ(4 + x).
- For y = 2eˣ(x³ – 3), the derivative is 2eˣ(x³ – 3) + 2eˣ·3x², or 2eˣ(x³ + 3x² – 3).
- For y = xeˣ, the nth derivative is eˣ(n + x).
The Quotient Rule
- The quotient rule states: (f/g)' = (f'g – fg')/g².
- Subtraction order in the numerator matters due to non-commutativity.
- For y = eˣ/(x + 1), derivative simplifies to xeˣ/(x + 1)².
- For y = (x + 2)/(3x² – x), derivative is [3x² – x – (x + 2)(6x – 1)]/(3x² – x)², further simplified.
- For y = [2 + xeˣ]/(x² – 2), use both rules: numerator's derivative involves product rule for xeˣ.
Application: Tangent Line Example
- To find the tangent line to y = 2x/(x + 1) at x = 2:
- Slope: y' = 2/(x + 1)², so at x = 2, slope = 2/9.
- Point: (2, 4/3). Equation: y – 4/3 = (2/9)(x – 2).
Working with Unknown Functions
- For y = x²f(x), derivative is 2xf(x) + x²f'(x).
- For y = √x + x·f(x)/eˣ, use product rule for x·f(x), then quotient rule for overall derivative.
Key Terms & Definitions
- Product Rule — The derivative of two multiplied functions f and g: (fg)' = f'g + fg'.
- Quotient Rule — The derivative of a quotient f/g: (f/g)' = (f'g – fg')/g².
- Differentiable Function — A function whose derivative exists for all points in its domain.
- nth Derivative — The result of differentiating a function n times.
- Point-Slope Form — Equation of a line: y – y₀ = m(x – x₀).
Action Items / Next Steps
- Memorize the product and quotient rule formulas.
- Practice differentiating products and quotients of functions.
- Review sections 3.5 and beyond for additional derivative rules.