Year 12 Differentiation (Power Rule) Worksheet — Level 3 (Set 2) — Answers
Answer sheet with worked solutions for the Year 12 calculus worksheet.
Year 12 · Mathematics · Calculus
Year 12 Differentiation (Power Rule) Worksheet — Level 3 (Set 2) — Answers
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Use the power rule: d/dx(axⁿ) = naxⁿ⁻¹.
1.Differentiate y = 2x6 − x3 + 7x2
Answer: dy/dx = 12x5 − 3x2 + 14x
- d/dx(2x6) = 12x5
- d/dx(−x3) = −3x2
- d/dx(7x2) = 14x
2.Differentiate y = −x5 + x − 1
Answer: dy/dx = −5x4 + 1
- d/dx(−x5) = −5x4
- d/dx(x) = 1
- The derivative of the constant −1 is 0
3.Differentiate y = 3x6 − 3x5 − x
Answer: dy/dx = 18x5 − 15x4 − 1
- d/dx(3x6) = 18x5
- d/dx(−3x5) = −15x4
- d/dx(−x) = −1
4.If f(x) = −4x6 − 2x5 + 7x3, find f′(2).
Answer: f′(2) = −844
- d/dx(−4x6) = −24x5
- d/dx(−2x5) = −10x4
- d/dx(7x3) = 21x2
- f′(x) = −24x5 − 10x4 + 21x2
- Substitute x = 2: −844
5.If f(x) = 4x5 + 7x4 − 2, find f′(−2).
Answer: f′(−2) = 96
- d/dx(4x5) = 20x4
- d/dx(7x4) = 28x3
- The derivative of the constant −2 is 0
- f′(x) = 20x4 + 28x3
- Substitute x = −2: 96
6.If f(x) = 8x6 + 4x5 − 4x4, find f′(−1).
Answer: f′(−1) = −12
- d/dx(8x6) = 48x5
- d/dx(4x5) = 20x4
- d/dx(−4x4) = −16x3
- f′(x) = 48x5 + 20x4 − 16x3
- Substitute x = −1: −12
7.If f(x) = 2x6 − 7x3 − 1, find f′(1).
Answer: f′(1) = −9
- d/dx(2x6) = 12x5
- d/dx(−7x3) = −21x2
- The derivative of the constant −1 is 0
- f′(x) = 12x5 − 21x2
- Substitute x = 1: −9
8.Differentiate y = 5x4 + 4x + 8
Answer: dy/dx = 20x3 + 4
- d/dx(5x4) = 20x3
- d/dx(4x) = 4
- The derivative of the constant 8 is 0
9.Differentiate y = −x6 − 3x4 − 7x3
Answer: dy/dx = −6x5 − 12x3 − 21x2
- d/dx(−x6) = −6x5
- d/dx(−3x4) = −12x3
- d/dx(−7x3) = −21x2
10.If f(x) = −2x5 − 3x4 + 3x2, find f′(1).
Answer: f′(1) = −16
- d/dx(−2x5) = −10x4
- d/dx(−3x4) = −12x3
- d/dx(3x2) = 6x
- f′(x) = −10x4 − 12x3 + 6x
- Substitute x = 1: −16
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