Rules of differentiation

NUST NET Maths · Maths · Differentiation. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

The derivative dy/dx is the rate at which y changes with x. In the test you never work from first principles: you recognise the shape of the function and apply the matching rule.

Ask three questions. Is it two functions multiplied (product rule)? Is it one divided by another (quotient rule)? Is it a function inside another function (chain rule)? For the chain rule, differentiate the outside, leave the inside alone, then multiply by the derivative of the inside.

When x and y are mixed in one equation, differentiate every term with respect to x and attach dy/dx to each y term. When x and y are both given in terms of t, divide the two derivatives.

Rules to remember

Product: (uv)′ = u′v + uv′. Quotient: (u/v)′ = (u′v − uv′)/v². Chain: dy/dx = (dy/du) × (du/dx)
Derivatives: sin x → cos x, cos x → −sin x, tan x → sec² x, cot x → −cosec² x, sec x → sec x tan x, cosec x → −cosec x cot x
e^ax → a e^ax; a^x → a^x ln a; ln x → 1/x; ln f(x) → f′(x)/f(x)
sin⁻¹ x → 1/√(1 − x²); cos⁻¹ x → −1/√(1 − x²); tan⁻¹ x → 1/(1 + x²)
Implicit f(x, y) = 0: dy/dx = −(derivative of f treating y as constant) ÷ (derivative of f treating x as constant)
Parametric: dy/dx = (dy/dt) ÷ (dx/dt), and d²y/dx² = [d/dt (dy/dx)] ÷ (dx/dt)
n^th derivative of e^ax is aⁿ e^ax; of sin ax it is aⁿ sin(ax + nπ/2)

Common mistake

Forgetting the derivative of the inside in the chain rule, for example writing d/dx (sin 3x) = cos 3x instead of 3 cos 3x. Every time the inside is not just x, multiply by its derivative.

Worked example

Find dy/dx at the point (1, 1) on the curve x² + 3xy − y² = 3.

Differentiate each term with respect to x: 2x + 3y + 3x(dy/dx) − 2y(dy/dx) = 0.
Collect dy/dx: (3x − 2y)(dy/dx) = −(2x + 3y).
Put x = 1, y = 1: (3 − 2)(dy/dx) = −(2 + 3), so dy/dx = −5/1.

Answer: dy/dx = −5

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