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Grade 12 Calculus Rules Summary

CAPS / NSC & IEB · Differential Calculus · Paper 1

→ Limits & Notation

Limit $\lim_{x \to a} f(x) = L$

The derivative, written $f'(x) \quad \dfrac{dy}{dx} \quad D_x[\,f(x)\,]$

A limit describes what the function approaches, not what it equals there. $\lim_{x\to 2}\dfrac{x^{2}-4}{x-2} = 4$ even though the function is undefined at $x = 2$ — simplify first, then substitute.

∂ First Principles

Definition $f'(x) = \lim_{h \to 0} \dfrac{f(x+h) - f(x)}{h}$

1 write $f(x+h)$ by substituting $x+h$ everywhere $x$ appears

2 subtract $f(x)$ and simplify the numerator

3 factor out $h$ and cancel it

4 only now let $h \to 0$

If the question says "from first principles", the rules earn nothing — the method is what is being marked. Keep $\lim_{h\to 0}$ written on every line until step 4; dropping it early costs marks even when the answer is right.

ƒ The Rules

Constant $\dfrac{d}{dx}(k) = 0$

Power rule $\dfrac{d}{dx}(x^{n}) = n\,x^{n-1}$

Constant multiple $\dfrac{d}{dx}\big(k\,f(x)\big) = k\,f'(x)$

Sum or difference $\dfrac{d}{dx}\big(f \pm g\big) = f' \pm g'$

Rewrite before you differentiate $\dfrac{1}{x^{n}} = x^{-n}$

$\sqrt[n]{x^{m}} = x^{\frac{m}{n}}$

CAPS has no product, quotient or chain rule. If an expression looks like it needs one, it is telling you to expand or divide through first. A fraction with a single term on the bottom splits into separate powers of $x$ — that is almost always the intended step.

∕ Gradients & Tangents

Average gradient $\dfrac{f(b) - f(a)}{b - a}$

Gradient at a point $m = f'(a)$

Tangent at $x=a$ $y - f(a) = f'(a)\,(x - a)$

Average gradient is between two points and needs no calculus; the gradient at a point is the derivative. The exam uses both words deliberately, and they are different questions.

∿ What the Derivatives Tell You

Condition	Meaning
$f'(x) > 0$	function increasing
$f'(x) < 0$	function decreasing
$f'(x) = 0$	stationary point
$f''(x) > 0$	concave up — a local minimum
$f''(x) < 0$	concave down — a local maximum
$f''(x) = 0$	possible point of inflection

$f'(x) = 0$ finds where the turning points are; $f''(x)$ tells you which kind. Answering "maximum" without justifying it from the second derivative or a sign table leaves marks on the table.

∫ Sketching a Cubic

General form $y = ax^{3} + bx^{2} + cx + d$

$y$-intercept $(0;\ d)$

$x$-intercepts solve $y=0$, usually via the factor theorem

Turning points $f'(x) = 0$

Point of inflection $x = -\dfrac{b}{3a}$

A cubic has exactly one point of inflection and it always sits halfway between the two turning points. If your two turning points are not symmetric about $-\frac{b}{3a}$, one of them is wrong.

⤒ Optimisation

1 write the quantity to be maximised or minimised as a formula

2 use the constraint to remove the second variable

3 differentiate and set the derivative to zero

4 answer the question that was asked

Step 2 is the whole difficulty: you cannot differentiate a formula in two variables. Step 4 matters more than it looks — the question often wants the dimensions, not the maximum volume, and the mark is for the thing asked for.

⏱ Rates of Change

Displacement $s(t)$

Velocity $v(t) = s'(t)$

Acceleration $a(t) = v'(t) = s''(t)$

At rest $v(t) = 0$

Maximum height $v(t) = 0$, then substitute back into $s(t)$

"Rate of change" always means differentiate with respect to the variable named. If the question says "per second", the derivative is with respect to $t$ even when the formula is written in another letter.

✓ Before You Move On

Rewrote first? every root and every fraction turned into a power of $x$ before differentiating

Kept the limit? $\lim_{h\to 0}$ written on every line of a first-principles answer until the last

Justified the type? said why a stationary point is a maximum or a minimum

Answered the question? gave the dimension, time or value actually asked for, with its unit

Calculus is the most learnable section in Paper 1.

It has few rules and a fixed set of question types, which is exactly why it rewards being taught properly rather than practised blindly. I tutor Grade 12 maths one-to-one in Gqeberha, at your own table, in English or Afrikaans. The first consultation is free.

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An independent revision aid aligned to the CAPS/NSC Mathematics syllabus. Not an official Department of Basic Education document.