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Calculus

Differential calculus

First principles, power rule, cubic graphs, stationary points, and optimisation.

CAPS

Grade 12 · Mathematics · Calculus

This sits on Mathematics Paper 1.

  • The paper

    Paper 1 is algebra, sequences and series, finance, functions, differential calculus and probability. If a formula is on the formula sheet, copy it before you substitute.

  • Cognitive level

    CAPS spreads the marks across knowledge, routine procedures, complex procedures and problem solving. A routine question still wants the line of working. A problem-solving question wants you to choose the method, not only to calculate.

  • The relation

    f'(x)=limh→0[f(x+h) − f(x)]h. First principles

  • How a marker reads it

    Show A'' or a first-derivative sign test to prove a maximum — required on NSC Paper 1.

Each Mathematics paper is 150 marks. A correct substitution can earn a method mark even when the last arithmetic slips, if the formula was the right one.

Diagram

Worked example

A rectangular kraal uses 40 m of fencing, one side against a barn. Maximise the area.

Before any working — hunt first

Given

Numbers and units already in the sentence: 40 m. Copy each one into a column. A name without a number is not given.

Missing

After you copy the given, the letter that still has no number is missing. The paper is pointing at: Maximise the area. Circle that word — that is the missing variable.

Find

We are required to find: Maximise the area. Write that as a symbol on the left, with a question mark, before you touch a formula.

Calculate

Write f'(x) = limh→0[f(x+h) − f(x)]h with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Maximise the area.

Simplified approach

Let width x, then length = 40 − 2x. Area A = x(40 − 2x). Differentiate, set A' = 0.

Culture note

This is the classic cattle-kraal optimisation — every Limpopo farm paper uses it.

Formulas

f'(x)=limh→0[f(x+h) − f(x)]h

First principles

ddx (axⁿ)=naxⁿ⁻¹

Power rule

How to hunt the missing letter

Do not start by naming the letters. First hunt the missing one: copy every number you can see (Given). The letter with no number is Missing. The asking word is Find. The formula that still contains that letter is what you Calculate.

How to hunt: (1) Underline every number and its unit — those are Given. (2) Circle the asking word — find, calculate, how far, what is. That word names Find. (3) The letter the asking word names, which has no number yet, is Missing. (4) Write the formula with empty letters. Cross out every given letter. The letter still standing is what you Calculate.

  • f'(x)

    derivative

    How you spot this in the sentence

    When the sentence names the derivative, that letter is f'(x). Copy the number next to it into Given. If the sentence never names it, f'(x) is Missing. The gradient of the tangent at x. Instantaneous rate of change.

    Example:If f(x) = x², f'(x) = 2x. At x = 3 the slope is 6.

  • h

    small step

    How you spot this in the sentence

    When the sentence names the small step, that letter is h. Copy the number next to it into Given. If the sentence never names it, h is Missing. The nudge in x used in first principles. Then let h → 0.

    Example:f'(x)=limh→0[f(x+h) − f(x)]h.

  • n

    exponent

    How you spot this in the sentence

    When the sentence names the exponent, that letter is n. Copy the number next to it into Given. If the sentence never names it, n is Missing. Power rule: d/dx (axⁿ) = n a xⁿ⁻¹.

    Example:ddx (4x³)=12x².

Exam tricks you must know

These are the moves markers look for — not shortcuts that skip the method, but the habits that earn the method mark.

  • Do not skip lim h→0 — it is the first method mark

    How:Write f'(x) = limh→0[f(x+h) − f(x)]h as the first line, with a division line, before you expand. Expand the numerator, cancel h, then bring the limit in. Dropping the lim notation on Gauteng and KZN Paper 1 docks the opening mark.

    When to use it:First principles of a derivative — any f(x) = x², x³, or a linear.

  • Always CAST — the second quadrant is a mark on its own

    How:Sketch CAST. Read the sign of sin, cos or tan from the quadrant, not from memory of a special angle. For a general solution write the acute angle first, then 180° − θ or 360° − θ. Leaving the second angle is the KZN and Gauteng dock.

    When to use it:Any sin / cos / tan equation that asks for θ in an interval.

  • Decide first: is money going in (F) or a loan being paid off (P)?

    How:Underline the asking word. “Future value”, “how much in the account”, “accumulate” → F. “Present value”, “loan”, “outstanding balance”, “how much can she borrow” → P. Write F or P in the Find cell before you open the formula sheet. The wrong letter loses the whole question.

    When to use it:Every compound-interest and annuity question on Paper 1.

Step 1 / 6

Area function

A(x)=40x−2x².

Diagram
Diagram
A=40x−2x²

Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.

4. Step-by-step working

  1. 01

    Area function

    A(x)=40x−2x².

    A=40x−2x²

  2. 02

    Differentiate

    A'=40−4x.

    A'=40−4x

  3. 03

    Stationary

    40−4x=0→x=10 m. Length=20 m. Max area=200 m².

    200 m²

  4. 04

    Step F — Box the answer with its unit

    x=10 m, length 20 m, Amax=200 m²

More worked examples

Board walkthroughs on this topic. Open one, copy the working, then try the next.

Example 1 / 20

Differentiate y = x³ with respect to x.

How we got there

Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.

Explain in another language · 11

1. What is the question asking?

Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Differentiate y = x³ with respect to x.

Before any working — hunt first

Given

Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.

Missing

After you copy the given, the letter that still has no number is missing. The paper is pointing at: Differentiate y = x³ with respect to x. Circle that word — that is the missing variable.

Find

We are required to find: Differentiate y = x³ with respect to x. Write that as a symbol on the left, with a question mark, before you touch a formula.

Calculate

Write 3x² with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Differentiate y = x³ with respect to x.

In other words — as if you have never seen this before

If you have never seen this before: Power rule: ddx [axⁿ] = a n xⁿ⁻¹. Constant → 0. We will name a method in words, write the formula with no numbers, then substitute one line at a time.

2. What do we already know?

y=x³.

3. Which method — and why this one?

Power rule: ddx [axⁿ]=a n xⁿ⁻¹. Constant→0.

4. Step-by-step working

  1. 01 / 04

    Differentiate

    dydx=3x².

    3x²

    Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.

  2. 02 / 04

    Sense-check

    Drop the power onto the coefficient, then subtract 1 from the power.

    3x²

    Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.

  3. 03 / 04

    Step F — Box the answer with its unit

    3x²

    Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.

  4. 04 / 04

    Step H — What the marker is looking for

    On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.

    Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.

5. The answer

3x²

Copy onto your answer sheet — in this order

Formula: 3x²

Answer: 3x²

6. Check it like a marker

The derivative of a constant is 0, not the constant.

7. A trap first-timers fall into

First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”

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