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: Find f'(x) from first principles if f(x) = x².
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: f'(x) from first principles if f(x) = x². Circle that word — that is the missing variable.
Find
We are required to find: f'(x) from first principles if f(x) = x². 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 f'(x) from first principles if f(x) = x².
In other words — as if you have never seen this before
If you have never seen this before: Write the definition, expand f(x+h) on the page, subtract, divide by h, then let h → 0. Do not jump to the answer. 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?
Given: f(x) = x². Unknown: f'(x) from first principles — not from the power-rule table.
3. Which method — and why this one?
Write the definition, expand f(x+h) on the page, subtract, divide by h, then let h → 0. Do not jump to the answer.
4. Step-by-step working
01 / 10
Write the definition
Copy this line first so the marker can see the method: f'(x) = limh→0[f(x+h) − f(x)]h.
f'(x)=limh→0[f(x+h) − f(x)]hWhy 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.
02 / 10
Write f(x)
The given function is f(x) = x². Leave it as it is.
f(x)=x²Why 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.
03 / 10
Write f(x+h)
Replace every x with (x+h). Do not expand yet — first write the substitution so a marker can tick the method mark.
f(x+h)=(x+h)²Why 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.
04 / 10
Expand the brackets
Now multiply out, one pair at a time, on the page: (x+h)² = x² + 2xh + h²
(x+h)²=x²+2xh+h²Why 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.
05 / 10
Subtract f(x)
f(x+h) − f(x) = 2xh + h². The original f(x) terms cancel. If they do not cancel, the expansion slipped.
2xh+h²Why 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.
06 / 10
Divide by h
Every term is divided by h. h is approaching 0, not equal to 0, so h ≠ 0 and we may cancel: (2xh + h²) / h = 2x + h
2x+hWhy 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.
07 / 10
Let h → 0
Now take the limit. The leftover h-terms die. What remains is f'(x) = 2x. Box it with the lim still written on the previous line.
f'(x)=2xWhy 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.
08 / 10
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
09 / 10
Step F — Box the answer with its unit
2x
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.
10 / 10
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
2x
Copy onto your answer sheet — in this order
Formula: f'(x)=limh→0[f(x+h) − f(x)]h
Answer: 2x
6. Check it like a marker
Put x back into the power rule as a check only: it should match 2x. The exam mark is for the seven lines above, not for quoting the table.
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…”