Differential Calculus Formula Sheet (Grade 12 CAPS)

First principles, derivative rules, tangents, stationary points and optimisation formulas for Grade 12 CAPS differential calculus with NSC exam notes.

Differential calculus is worth about 35 marks in NSC Paper 1 and is highly predictable: first principles, rules of differentiation, cubic graph sketching, and rate of change or optimisation.

Formulas

Formula	Expression	Note
Derivative from first principles	`f′(x) = lim(h→0) [f(x + h) − f(x)]/h`	The limit notation must be written on every line until h is cancelled.
Power rule	`d/dx [axⁿ] = anxⁿ⁻¹`	Rewrite surds and fractions as powers of x before differentiating.
Tangent to a curve	`y − y₁ = f′(x₁)(x − x₁)`	f′(x₁) is the gradient at the point of contact.
Stationary points	`f′(x) = 0`	Solve for x, then substitute into f(x) for the y-coordinate.
Point of inflection	`f″(x) = 0`	For a cubic it is the average of the two stationary point x-values.
Rate of change	`Rate = f′(t)`	Velocity is s′(t) and acceleration is s″(t).
Sum and difference rule	`d/dx [f(x) ± g(x)] = f′(x) ± g′(x)`	Differentiate term by term; expand brackets and simplify fractions first.
Derivative of a constant	`d/dx [c] = 0`	A constant term disappears when you differentiate.
Average gradient	`[f(b) − f(a)] ÷ (b − a)`	The gradient of the straight line between two points on the curve.
Second derivative	`f″(x) = d/dx [f′(x)]`	f″(x) > 0 means concave up; f″(x) < 0 means concave down.

How to use them

Use second-derivative or sign testing to classify maximum versus minimum.
Optimisation questions need a constraint equation before differentiating.
Never use the power rule inside a first-principles question — no marks are awarded.
Worked example: if f(x) = 2x³ − 3x², then f′(x) = 6x² − 6x = 6x(x − 1), so the stationary points are at x = 0 and x = 1. f″(x) = 12x − 6 shows that (0; 0) is a local maximum and (1; −1) a local minimum.

Frequently asked questions

Are all Differential Calculus formulas given in the NSC exam?

Some are printed on the official information sheet, but not all. Treat every formula on this page as examinable and learn it with a worked example so you recognise when to use it.

Which NSC paper tests Differential Calculus?

Differential Calculus is examined in Paper 1 of the NSC Mathematics examination.

How do I practise these formulas?

Snap a differential calculus question into the Snap&Learn AI solver — it shows the substitution and every method mark the way the NSC memo does.

Snap&Learn gives South African Matric learners instant, CAPS-aligned, step-by-step Mathematics solutions. Snap or upload a question and the AI shows every method mark the way the NSC memo awards them. Your first three AI solutions are free.