Differentiation: The Rules and How Exams Use Them — Worksheet & Memo

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Grade 12 · Calculus

Differentiation: The Rules and How Exams Use Them

MEMORANDUM

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/ 37

Answer all questions. Show all your working — marks are awarded for method as well as the final answer. Teacher copy: accept any correct equivalent method.

1
Differentiate y = x⁴. (power rule)
(1)
4x³✓ (1)
2
Differentiate y = 5x³ − 2x² + 7. (polynomial)
(2)
15x² − 4x✓✓ (2)
3
Differentiate y = 2x² − 3x + 9. (polynomial)
(2)
4x − 3✓✓ (2)
4
Differentiate y = 3/x (write as 3x⁻¹). (rewrite)
(2)
−3x⁻² = (−3)/x²✓✓ (2)
5
Differentiate y = √x (write as x^1/2). (rewrite)
(2)
1/2x^(−1/2) = 1/(2√x)✓✓ (2)
6
From first principles, differentiate f(x) = 3x. (first principles)
(4)
f′(x) = lim_h→0 (3(x + h) − 3x)/h = lim_h→0 3h/h = lim_h→0 3 = 3✓✓✓✓ (4)
7
Gradient of y = x² + 2x at x = 1. (gradient)
(2)
2x + 2 at x=1 = 4✓✓ (2)
8
Gradient of y = x³ at x = 2. (gradient)
(2)
3x² at x=2 = 12✓✓ (2)
9
Find x where y = x² − 6x has a turning point. (turning point)
(2)
2x − 6 = 0 → x = 3✓✓ (2)
10
Find the turning points of y = x³ − 12x. (turning point)
(4)
3x² − 12=0 → x = ±2 → (2 ; −16) and (−2 ; 16)✓✓✓✓ (4)
11
Determine f′(x) from first principles if f(x) = 2x² − 3x.
(5)
f(x + h) = 2(x + h)² − 3(x + h) = 2x² + 4xh + 2h² − 3x − 3h ✓; f(x + h) − f(x) = 4xh + 2h² − 3h ✓; ÷ h: 4x + 2h − 3 ✓; lim as h→0 ✓; f′(x) = 4x − 3 ✓(5)
12
Determine f′(x) from first principles if f(x) = 3/x.
(5)
f′(x) = lim_h→0 (3/(x + h) − 3/x)/h ✓
= lim_h→0 (3x − 3(x + h))/(hx(x + h)) = lim_h→0 −3h/(hx(x + h)) ✓✓
= lim_h→0 −3/(x(x + h)) ✓
= −3/x² ✓(5)
13
Use first principles to show that the derivative of f(x) = 7 is 0.
(2)
f(x + h) − f(x) = 7 − 7 = 0 ✓; 0 ÷ h = 0, so f′(x) = 0 ✓(2)
14
Explain why you may NOT substitute h = 0 at the start of a first-principles question.
(2)
It gives 0 ÷ 0, which is undefined ✓; the h in the denominator must be cancelled by simplifying first ✓(2)

TOTAL: 37 marks