Definition of Product Rule
Concept
Rule to differentiate product of two differentiable functions. Ensures correct derivative of f(x)·g(x). Fundamental in single-variable calculus.
Purpose
Computes derivative when direct differentiation not possible. Avoids error of simply multiplying derivatives.
Scope
Applies to any two functions with existing derivatives on interval. Extends to vector-valued functions.
Derivation of Product Rule
Limit Definition Basis
Starts from definition of derivative as limit of difference quotient. Uses f(x+h)g(x+h) - f(x)g(x).
Algebraic Manipulation
Adds and subtracts f(x+h)g(x) to restructure difference. Groups increments to separate changes in each function.
Final Limit Form
Limits of each grouped term yield f'(x)g(x) + f(x)g'(x). Demonstrates sum of products of derivatives and original functions.
Derivative of f(x)g(x):lim(h→0) [f(x+h)g(x+h) - f(x)g(x)] / h= 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]= f(x)g'(x) + f'(x)g(x) Formula and Notation
Standard Formula
d/dx [f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x)
Leibniz Notation
d(uv)/dx = u(dv/dx) + v(du/dx), where u=f(x), v=g(x)
Alternative Forms
Can be written as (fg)' = f'g + fg' or expanded for multiple functions by iterative application.
| Notation | Formula |
|---|---|
| Leibniz | d(uv)/dx = u(dv/dx) + v(du/dx) |
| Prime | (fg)' = f'g + fg' |
Worked Examples
Example 1: Polynomial Functions
f(x)=x², g(x)=3x+1
Derivative: f'(x)=2x, g'(x)=3
Result: d/dx [x²(3x+1)] = 2x(3x+1) + x²(3) = 6x² + 2x + 3x² = 9x² + 2x
Example 2: Exponential and Trigonometric
f(x)=eˣ, g(x)=sin x
Derivative: f'(x)=eˣ, g'(x)=cos x
Result: d/dx [eˣ sin x] = eˣ sin x + eˣ cos x = eˣ (sin x + cos x)
Given f(x)=x², g(x)=3x+1f'(x)=2xg'(x)=3d/dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)= 2x(3x+1) + x²(3)= 6x² + 2x + 3x²= 9x² + 2x Example 3: Logarithmic and Polynomial
f(x)=ln x, g(x)=x³
Derivative: f'(x)=1/x, g'(x)=3x²
Result: d/dx [ln x · x³] = (1/x)·x³ + ln x · 3x² = x² + 3x² ln x
Applications in Calculus
Function Differentiation
Essential for differentiating products in algebraic, trigonometric, exponential, logarithmic functions.
Physics and Engineering
Used to find rates where quantities are products of changing variables: velocity·mass, force·distance.
Economics and Biology
Models growth rates of products: population multiplied by resource consumption rate.
Comparison with Other Rules
Product Rule vs Chain Rule
Product rule: derivative of product of functions. Chain rule: derivative of composite functions.
Product Rule vs Quotient Rule
Quotient rule derives ratio of functions; product rule derives product.
Combined Use
Often combined: differentiate product of composite functions requires both rules.
Higher-Order Derivatives
Second Derivative
Use product rule twice or generalized Leibniz formula for nth derivative.
Leibniz Formula
Generalization: (fg)⁽ⁿ⁾ = Σ (k=0 to n) (n choose k) f⁽ᵏ⁾ g⁽ⁿ⁻ᵏ⁾
Second derivative:d²/dx² (fg) = f''g + 2f'g' + fg'' Applications
Used in Taylor series, differential equations, advanced calculus problems.
Common Mistakes
Ignoring Product Rule
Incorrectly differentiating product as product of derivatives yields false results.
Swapping Terms
Forgetting order: derivative of first times second plus first times derivative of second.
Applying to Non-Differentiable Functions
Rule requires differentiability; misuse leads to undefined expressions.
Extensions and Generalizations
Multiple Functions
Product rule extends to three or more functions by repeated application.
Vector-Valued Functions
Applies to dot and cross products with adapted product rules.
Multivariable Calculus
Partial derivatives use product rule in several variables context.
Visualization and Interpretation
Geometric Meaning
Derivative of area under curve formed by product of two functions’ values.
Graphical Representation
Slope of product function is sum of slopes weighted by opposite function values.
Rate of Change
Captures combined instantaneous rate of change of multiplied quantities.
| Function | Value at x | Derivative at x |
|---|---|---|
| f(x) | f(x₀) | f'(x₀) |
| g(x) | g(x₀) | g'(x₀) |
| (fg)(x) | f(x₀)·g(x₀) | f'(x₀)g(x₀) + f(x₀)g'(x₀) |
Practice Problems
Problem 1
Find d/dx of (x³)(cos x).
Problem 2
Differentiate (ln x)(x² + 1).
Problem 3
Compute derivative of (eˣ)(tan x).
Problem 4
Find second derivative of (x²)(sin x).
Solutions
Use product rule directly; verify results with stepwise differentiation.
References
- Stewart, James. Calculus: Early Transcendentals. Cengage Learning, Vol. 8, 2015, pp. 165-170.
- Thomas, George B., and Ross L. Finney. Calculus and Analytic Geometry. Addison-Wesley, Vol. 9, 2002, pp. 120-125.
- Spivak, Michael. Calculus. Publish or Perish, Vol. 4, 2008, pp. 65-70.
- Apostol, Tom M. Mathematical Analysis. Addison-Wesley, Vol. 2, 1974, pp. 98-102.
- Larson, Ron, and Bruce Edwards. Calculus. Brooks Cole, Vol. 10, 2013, pp. 145-150.