Differentiation
What this domain covers
Differentiation questions test the gradient as a limit, standard and composite derivatives, tangents and normals, stationary points and their classification, intervals where a function increases or decreases, and connected rates of change.
Core rules
- Standard results: d/dx(x^n) = nx^(n−1); d/dx(sin x) = cos x; d/dx(cos x) = −sin x; d/dx(e^x) = e^x; d/dx(ln x) = 1/x.
- Chain rule: dy/dx = dy/du · du/dx. Product rule: (uv)′ = u′v + uv′. Quotient rule: (u/v)′ = (u′v − uv′)/v².
- Tangent at x = a: gradient m = f′(a), equation y − f(a) = m(x − a). Normal gradient is −1/m.
- Stationary points: solve f′(x) = 0, then classify with f″ (negative gives a maximum, positive a minimum) or a sign table.
- Connected rates: dV/dt = dV/dr · dr/dt.
Worked example
y = x³ − 3x² + 4. Then dy/dx = 3x² − 6x = 3x(x − 2), so stationary points occur at x = 0 and x = 2. Here d²y/dx² = 6x − 6, which is −6 at x = 0 (maximum, y = 4) and +6 at x = 2 (minimum, y = 0). The normal at x = 1: the gradient there is 3 − 6 = −3, so the normal has gradient ⅓ and passes through (1, 2), giving y − 2 = ⅓(x − 1).
Traps
- Omitting the inner derivative when differentiating (2x + 5)^4 or sin 3x.
- Quoting the tangent's gradient for the normal, or forgetting −1/m entirely.
- Declaring a point of inflexion whenever f″ = 0; the sign of f′ still has to be examined.
- Forgetting to differentiate both variables in a connected-rates chain.