Grade 12 Maths · Calculus

Differentiate from first principles

Find the derivative using the limit definition, working out f(x + h) − f(x), dividing by h and letting h go to 0.

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Question 8.1 · May/June 2025 · Paper 1 · National (DBE) · 5 marks

8.1 Determine

f^{'} (x)

from first principles if it is given that

f (x) = x^{2} - 2

.(5)

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More past paper questions: differentiate from first principles

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3. Question 8.3.1 · November 2024 · Paper 1 · National (DBE) · 5 marks
8.3 Given: $f (x) = - 6 x^{2}$
8.3.1 Determine $f^{'} (x)$ from first principles.(5)
Show the memo (4 ticks)Memo
1. ✓
  Substitute into definition $f^{'} (x) = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h}$
  $= lim_{h \to 0} \frac{- 6 ( x + h ) ^{2} - ( - 6 x ^{2} )}{h}$
  1 mark · Differentiate from first principles
2. ✓
  Expand and simplify numerator $f (x + h) = - 6 (x^{2} + 2 x h + h^{2}) = - 6 x^{2} - 12 x h - 6 h^{2}$
  Numerator: $- 6 x^{2} - 12 x h - 6 h^{2} + 6 x^{2} = - 12 x h - 6 h^{2}$
  1 mark · Differentiate from first principles
3. ✓
  Factorise h $= lim_{h \to 0} \frac{h ( - 12 x - 6 h )}{h}$
  2 marks · Differentiate from first principles
4. ✓
  Calculate limit $= - 12 x$
  1 mark · Differentiate from first principles
4. Question 8.1 · September 2024 · Paper 1 · Provincial · 4 marks
8.1 If $f (x) = - 2 x^{2} + 3 x$ , determine $f^{'} (x)$ from first principles.(4)
Show the memo (4 ticks)Memo
1. ✓
  Substitute x+h $- 2 (x + h)^{2} + 3 (x + h)$
  1 mark · Differentiate from first principles
2. ✓
  SimplificationNumerator simplifies to $- 4 x h - 2 h^{2} + 3 h$
  1 mark · Differentiate from first principles
3. ✓
  Factorise and limit $lim_{h \to 0} (- 4 x - 2 h + 3)$
  1 mark · Differentiate from first principles
4. ✓
  Answer $- 4 x + 3$
  1 mark · Differentiate from first principles

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