Differentiation Techniques: Chain, Product and Quotient Rules
Differentiating more complex functions
Beyond the power rule, three techniques let you differentiate composite, product and quotient functions.
The chain rule
For a function of a function y = f(g(x)):
dy/dx = (dy/du) × (du/dx)Substitute u for the "inside" function.
- Example:
y = (3x² + 1)⁵. Let u = 3x² + 1, so y = u⁵. - dy/du = 5u⁴, du/dx = 6x.
- dy/dx = 5u⁴ × 6x = 30x(3x² + 1)⁴.
Useful results from the chain rule: d/dx[eᶠ⁽ˣ⁾] = f′(x)eᶠ⁽ˣ⁾, d/dx[ln f(x)] = f′(x)/f(x).
The product rule
For a product y = uv (two functions multiplied):
dy/dx = u(dv/dx) + v(du/dx)- Example:
y = x² sin x. u = x² (u′ = 2x), v = sin x (v′ = cos x). - dy/dx = x² cos x + sin x (2x) = x² cos x + 2x sin x.
The quotient rule
For a quotient y = u/v:
dy/dx = [v(du/dx) − u(dv/dx)] ÷ v²- Example:
y = x / (x + 1). u = x (u′ = 1), v = x + 1 (v′ = 1). - dy/dx = [(x+1)(1) − x(1)] ÷ (x+1)² = 1 ÷ (x+1)².
Choosing the right rule
- Chain: one function inside another (brackets to a power, e^(something), ln(something)).
- Product: two functions multiplied.
- Quotient: one function divided by another (or rewrite as a product with a negative power).
Implicit and parametric (brief)
- Implicit: differentiate each term with respect to x, using dy/dx when differentiating y terms.
- Parametric: if x and y are given in terms of t,
dy/dx = (dy/dt) ÷ (dx/dt).
Worked example
Differentiate y = (2x + 3)⁴ using the chain rule.
- u = 2x + 3, y = u⁴. dy/du = 4u³, du/dx = 2.
- dy/dx = 4u³ × 2 = 8(2x + 3)³. ✓
Common mistakes
- Forgetting to multiply by the derivative of the inside function (chain rule).
- Mixing up the product and quotient rule signs (quotient is v u′ minus u v′).
- Not dividing by v² in the quotient rule.
Exam tips
- Identify the structure first (composite / product / quotient) to pick the rule.
- Write out u, v (or u, du/dx) clearly before substituting.
- Simplify/factorise the final answer.
Key facts to remember
- Chain rule: dy/dx = (dy/du)(du/dx) — for a function inside a function.
- Product rule: (uv)′ = u v′ + v u′; Quotient rule: (u/v)′ = (v u′ − u v′)/v².
- Standard results: d/dx[eᶠ] = f′eᶠ, d/dx[ln f] = f′/f.