First principles
Keep the limit notation on every line. Drop it and you drop marks.
Cubic graph sketches
Find f'(x) = 0 for stationary points; use f''(x) to classify; find intercepts.
First principles, worked
For f(x) = 2x² − 1: f(x + h) − f(x) = 4xh + 2h². Dividing by h gives 4x + 2h, and letting h → 0 gives f′(x) = 4x.
A cubic, worked
For f(x) = x³ − 3x² − 9x + 5: f′(x) = 3x² − 6x − 9 = 3(x − 3)(x + 1), so the stationary points are at x = 3 and x = −1. f(3) = −22 is a local minimum and f(−1) = 10 is a local maximum. The point of inflection is halfway between them, at x = 1.
Rate of change, worked
If s(t) = 2t³ − 3t² metres, the velocity is s′(t) = 6t² − 6t, so at t = 2 seconds it is 6(4) − 6(2) = 12 m/s.
Tangents to a curve
A tangent question gives a curve and an x-value (or a gradient). The derivative gives the gradient, the original function gives the point, and y − y₁ = m(x − x₁) gives the line. Example: for f(x) = x³ − 2x at x = 1, f(1) = −1 and f′(x) = 3x² − 2, so m = f′(1) = 1 and the tangent is y = x − 2.
Optimisation, worked
An open box is made from a 12 cm × 12 cm square of cardboard by cutting a square of side x from each corner and folding up the sides. Its volume is V(x) = x(12 − 2x)². Then V′(x) = (12 − 2x)² − 4x(12 − 2x) = (12 − 2x)(12 − 6x). V′(x) = 0 gives x = 6, which makes a box with no base, so reject it, or x = 2. The maximum volume is V(2) = 2 × 8² = 128 cm³.
Reading a graph of f′(x)
- Where f′(x) = 0, f has a stationary point.
- Where f′(x) > 0, f is increasing; where f′(x) < 0, f is decreasing.
- For a cubic f, the graph of f′ is a parabola, and its turning point gives the x-coordinate of f's point of inflection.
- f is concave up where f″(x) > 0 and concave down where f″(x) < 0.
A checklist for every calculus question
- Rewrite roots and fractions as powers of x before differentiating.
- Use correct notation: f′(x) =, dy/dx = or Dₓ[…].
- Substitute into f(x), not f′(x), to find y-coordinates.
- State the nature of every stationary point, with a reason.
Optimisation
Always two equations: the quantity to maximise/minimise and a constraint. Substitute, differentiate, set to zero. When a question stops you, snap it into Snap&Learn — you'll get a CAPS-aligned step-by-step explanation in seconds, and your first three AI solutions are free.