Year 12 Differentiation (Power Rule) Worksheet — Level 3 — Answers
Answer sheet with worked solutions for the Year 12 calculus worksheet.
Year 12 · Mathematics · Calculus
Year 12 Differentiation (Power Rule) Worksheet — Level 3 — Answers
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Use the power rule: d/dx(axⁿ) = naxⁿ⁻¹.
1.Differentiate y = −2x6 + 8x5 − 2
Answer: dy/dx = −12x5 + 40x4
- d/dx(−2x6) = −12x5
- d/dx(8x5) = 40x4
- The derivative of the constant −2 is 0
2.If f(x) = 7x6 + 9x5 − 5x4, find f′(2).
Answer: f′(2) = 1904
- d/dx(7x6) = 42x5
- d/dx(9x5) = 45x4
- d/dx(−5x4) = −20x3
- f′(x) = 42x5 + 45x4 − 20x3
- Substitute x = 2: 1904
3.If f(x) = 8x2 − x − 7, find f′(−2).
Answer: f′(−2) = −33
- d/dx(8x2) = 16x
- d/dx(−x) = −1
- The derivative of the constant −7 is 0
- f′(x) = 16x − 1
- Substitute x = −2: −33
4.If f(x) = 7x6 + 4x4 − 4x3, find f′(2).
Answer: f′(2) = 1424
- d/dx(7x6) = 42x5
- d/dx(4x4) = 16x3
- d/dx(−4x3) = −12x2
- f′(x) = 42x5 + 16x3 − 12x2
- Substitute x = 2: 1424
5.If f(x) = 6x6 − 2x3 + 7, find f′(−1).
Answer: f′(−1) = −42
- d/dx(6x6) = 36x5
- d/dx(−2x3) = −6x2
- The derivative of the constant 7 is 0
- f′(x) = 36x5 − 6x2
- Substitute x = −1: −42
6.If f(x) = 7x6 + 3x5 + 9x4, find f′(−1).
Answer: f′(−1) = −63
- d/dx(7x6) = 42x5
- d/dx(3x5) = 15x4
- d/dx(9x4) = 36x3
- f′(x) = 42x5 + 15x4 + 36x3
- Substitute x = −1: −63
7.Differentiate y = −4x6 + 3x2 + 9
Answer: dy/dx = −24x5 + 6x
- d/dx(−4x6) = −24x5
- d/dx(3x2) = 6x
- The derivative of the constant 9 is 0
8.Differentiate y = x4 + 6x2 − 5
Answer: dy/dx = 4x3 + 12x
- d/dx(x4) = 4x3
- d/dx(6x2) = 12x
- The derivative of the constant −5 is 0
9.Differentiate y = 2x5 − 6x4 − 5x3
Answer: dy/dx = 10x4 − 24x3 − 15x2
- d/dx(2x5) = 10x4
- d/dx(−6x4) = −24x3
- d/dx(−5x3) = −15x2
10.Differentiate y = −5x6 + 9x2 − 4
Answer: dy/dx = −30x5 + 18x
- d/dx(−5x6) = −30x5
- d/dx(9x2) = 18x
- The derivative of the constant −4 is 0
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