Rules of Differentiation: Sum, Product and Quotient Laws

Computing every derivative from first principles would be exhausting. The rules of differentiation are shortcuts for the three ways functions combine: sums, products, and quotients. The sum rule differentiates term by term, the product rule handles multiplied functions, and the quotient rule handles division.

These rules matter because real formulas are almost always combinations — revenue is price times quantity, kinetic energy is a constant times velocity squared, and rational models divide one polynomial by another. With the three rules, you differentiate the pieces you know and assemble the answer mechanically.

Key Concepts

• Sum rule: the derivative of a sum is the sum of the derivatives. Differentiate each term separately.
• Product rule: for y = f(x)·g(x), the derivative is f'g + fg' — derivative of the first times the second, plus the first times the derivative of the second.
• Quotient rule: for y = f(x)/g(x), the derivative is (f'g − fg')/g² — same pattern as the product rule, but subtract, and divide by the denominator squared.
• Order matters: in the quotient rule, which function is f and which is g changes the sign of the answer.

Formulas

• Sum/difference: (f ± g)' = f' ± g'.
• Product: (f · g)' = f' · g + f · g'.
• Quotient: (f/g)' = (f' · g − f · g²)/g², valid where g(x) ≠ 0.
• Memory aid: "first-derivative times second, plus first times second-derivative" for products; "low d-high minus high d-low over low squared" for quotients.

Worked Example

Differentiate y = (x² + 1)(3x − 2) using the product rule, and evaluate y'(1).

Identify the factors: f(x) = x² + 1 with f'(x) = 2x, and g(x) = 3x − 2 with g'(x) = 3. Apply the product rule: y' = f'g + fg' = 2x(3x − 2) + (x² + 1)(3). Expand: 2x(3x − 2) = 6x² − 4x, and 3(x² + 1) = 3x² + 3. Adding gives y' = 6x² − 4x + 3x² + 3 = 9x² − 4x + 3. At x = 1: y'(1) = 9 − 4 + 3 = 8. As a cross-check, expand first: (x² + 1)(3x − 2) = 3x³ − 2x² + 3x − 2, whose term-by-term derivative is 9x² − 4x + 3 — the same result, and 9 − 4 + 3 = 8 at x = 1.

Common Misconceptions

• The derivative of a product is the product of the derivatives. It is not: (fg)' = f'g + fg', never f'g'.
• Quotient rule order does not matter. It does — swapping f and g flips the sign of the numerator.
• A sum of many terms needs the product rule. No: sums differentiate term by term. Only actual multiplication needs the product rule.

Real-World Applications

• Revenue: revenue = price × quantity is a product, differentiated with the product rule for marginal analysis.
• Rational models: efficiency ratios and average-cost functions need the quotient rule.
• Physics: momentum p = mv with varying mass and velocity differentiates as a product.
• Engineering: transfer functions that are ratios of polynomials are differentiated with the quotient rule.

In summary, three rules cover all combinations: sums split term by term, products follow f'g + fg', and quotients follow (f'g − fg')/g². Identify the structure, differentiate the pieces, and assemble — no first-principles algebra required.