MathsAges 16-18

Chain rule differentiation

Apply the chain rule to differentiate composite power functions f(x) = (ax + b)ⁿ and evaluate at a given x. Review suitability for your learners before teaching.

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Worksheet StudioPractice pack · Ages 16-18

Chain rule differentiation

Take your time. Show your thinking.

1. Using the chain rule, differentiate f(x) = (6x + 1)^3 and evaluate f′(2).
f(x) = (6x + 1)^3 · f′(2) at u = 13
2. Using the chain rule, differentiate f(x) = (6x + 3)^3 and evaluate f′(2).
f(x) = (6x + 3)^3 · f′(2) at u = 15
3. Using the chain rule, differentiate f(x) = (6x + 7)^4 and evaluate f′(1).
f(x) = (6x + 7)^4 · f′(1) at u = 13
4. Using the chain rule, differentiate f(x) = (2x + 2)^2 and evaluate f′(1).
f(x) = (2x + 2)^2 · f′(1) at u = 4
5. Using the chain rule, differentiate f(x) = (4x + 4)^2 and evaluate f′(2).
f(x) = (4x + 4)^2 · f′(2) at u = 12
6. Using the chain rule, differentiate f(x) = (6x + 6)^4 and evaluate f′(3).
f(x) = (6x + 6)^4 · f′(3) at u = 24
7. Using the chain rule, differentiate f(x) = (3x + 4)^4 and evaluate f′(1).
f(x) = (3x + 4)^4 · f′(1) at u = 7
8. Using the chain rule, differentiate f(x) = (4x + 4)^3 and evaluate f′(4).
f(x) = (4x + 4)^3 · f′(4) at u = 20
9. Using the chain rule, differentiate f(x) = (6x + 3)^3 and evaluate f′(1).
f(x) = (6x + 3)^3 · f′(1) at u = 9
10. Using the chain rule, differentiate f(x) = (2x + 1)^3 and evaluate f′(3).
f(x) = (2x + 1)^3 · f′(3) at u = 7
11. Using the chain rule, differentiate f(x) = (2x + 3)^2 and evaluate f′(3).
f(x) = (2x + 3)^2 · f′(3) at u = 9
12. Using the chain rule, differentiate f(x) = (4x + 3)^4 and evaluate f′(2).
f(x) = (4x + 3)^4 · f′(2) at u = 11

12 questions · Seed 47 · Age guidance, not curriculum certification

View the answer key

Worksheet StudioAnswer key · Ages 16-18

Chain rule differentiation

Answers and reasoning. Review before teaching.

1. Using the chain rule, differentiate f(x) = (6x + 1)^3 and evaluate f′(2).
3042
By the chain rule, f′(x) = 3 × 6(6x + 1)^2 = 18(6x + 1)^2. At x = 2, f′(2) = 18 × 13^2 = 3042.
2. Using the chain rule, differentiate f(x) = (6x + 3)^3 and evaluate f′(2).
4050
By the chain rule, f′(x) = 3 × 6(6x + 3)^2 = 18(6x + 3)^2. At x = 2, f′(2) = 18 × 15^2 = 4050.
3. Using the chain rule, differentiate f(x) = (6x + 7)^4 and evaluate f′(1).
52728
By the chain rule, f′(x) = 4 × 6(6x + 7)^3 = 24(6x + 7)^3. At x = 1, f′(1) = 24 × 13^3 = 52728.
4. Using the chain rule, differentiate f(x) = (2x + 2)^2 and evaluate f′(1).
16
By the chain rule, f′(x) = 2 × 2(2x + 2)^1 = 4(2x + 2)^1. At x = 1, f′(1) = 4 × 4^1 = 16.
5. Using the chain rule, differentiate f(x) = (4x + 4)^2 and evaluate f′(2).
96
By the chain rule, f′(x) = 2 × 4(4x + 4)^1 = 8(4x + 4)^1. At x = 2, f′(2) = 8 × 12^1 = 96.
6. Using the chain rule, differentiate f(x) = (6x + 6)^4 and evaluate f′(3).
331776
By the chain rule, f′(x) = 4 × 6(6x + 6)^3 = 24(6x + 6)^3. At x = 3, f′(3) = 24 × 24^3 = 331776.
7. Using the chain rule, differentiate f(x) = (3x + 4)^4 and evaluate f′(1).
4116
By the chain rule, f′(x) = 4 × 3(3x + 4)^3 = 12(3x + 4)^3. At x = 1, f′(1) = 12 × 7^3 = 4116.
8. Using the chain rule, differentiate f(x) = (4x + 4)^3 and evaluate f′(4).
4800
By the chain rule, f′(x) = 3 × 4(4x + 4)^2 = 12(4x + 4)^2. At x = 4, f′(4) = 12 × 20^2 = 4800.
9. Using the chain rule, differentiate f(x) = (6x + 3)^3 and evaluate f′(1).
1458
By the chain rule, f′(x) = 3 × 6(6x + 3)^2 = 18(6x + 3)^2. At x = 1, f′(1) = 18 × 9^2 = 1458.
10. Using the chain rule, differentiate f(x) = (2x + 1)^3 and evaluate f′(3).
294
By the chain rule, f′(x) = 3 × 2(2x + 1)^2 = 6(2x + 1)^2. At x = 3, f′(3) = 6 × 7^2 = 294.
11. Using the chain rule, differentiate f(x) = (2x + 3)^2 and evaluate f′(3).
36
By the chain rule, f′(x) = 2 × 2(2x + 3)^1 = 4(2x + 3)^1. At x = 3, f′(3) = 4 × 9^1 = 36.
12. Using the chain rule, differentiate f(x) = (4x + 3)^4 and evaluate f′(2).
21296
By the chain rule, f′(x) = 4 × 4(4x + 3)^3 = 16(4x + 3)^3. At x = 2, f′(2) = 16 × 11^3 = 21296.

12 questions · Seed 47 · Age guidance, not curriculum certification

Worksheet StudioPractice pack · Ages 16-18

Chain rule differentiation

Take your time. Show your thinking.

1. Using the chain rule, differentiate f(x) = (6x + 1)^3 and evaluate f′(2).
f(x) = (6x + 1)^3 · f′(2) at u = 13
2. Using the chain rule, differentiate f(x) = (6x + 3)^3 and evaluate f′(2).
f(x) = (6x + 3)^3 · f′(2) at u = 15
3. Using the chain rule, differentiate f(x) = (6x + 7)^4 and evaluate f′(1).
f(x) = (6x + 7)^4 · f′(1) at u = 13
4. Using the chain rule, differentiate f(x) = (2x + 2)^2 and evaluate f′(1).
f(x) = (2x + 2)^2 · f′(1) at u = 4
5. Using the chain rule, differentiate f(x) = (4x + 4)^2 and evaluate f′(2).
f(x) = (4x + 4)^2 · f′(2) at u = 12
6. Using the chain rule, differentiate f(x) = (6x + 6)^4 and evaluate f′(3).
f(x) = (6x + 6)^4 · f′(3) at u = 24
7. Using the chain rule, differentiate f(x) = (3x + 4)^4 and evaluate f′(1).
f(x) = (3x + 4)^4 · f′(1) at u = 7
8. Using the chain rule, differentiate f(x) = (4x + 4)^3 and evaluate f′(4).
f(x) = (4x + 4)^3 · f′(4) at u = 20
9. Using the chain rule, differentiate f(x) = (6x + 3)^3 and evaluate f′(1).
f(x) = (6x + 3)^3 · f′(1) at u = 9
10. Using the chain rule, differentiate f(x) = (2x + 1)^3 and evaluate f′(3).
f(x) = (2x + 1)^3 · f′(3) at u = 7
11. Using the chain rule, differentiate f(x) = (2x + 3)^2 and evaluate f′(3).
f(x) = (2x + 3)^2 · f′(3) at u = 9
12. Using the chain rule, differentiate f(x) = (4x + 3)^4 and evaluate f′(2).
f(x) = (4x + 3)^4 · f′(2) at u = 11

12 questions · Seed 47 · Age guidance, not curriculum certification

Worksheet StudioAnswer key · Ages 16-18

Chain rule differentiation

Answers and reasoning. Review before teaching.

1. Using the chain rule, differentiate f(x) = (6x + 1)^3 and evaluate f′(2).
3042
By the chain rule, f′(x) = 3 × 6(6x + 1)^2 = 18(6x + 1)^2. At x = 2, f′(2) = 18 × 13^2 = 3042.
2. Using the chain rule, differentiate f(x) = (6x + 3)^3 and evaluate f′(2).
4050
By the chain rule, f′(x) = 3 × 6(6x + 3)^2 = 18(6x + 3)^2. At x = 2, f′(2) = 18 × 15^2 = 4050.
3. Using the chain rule, differentiate f(x) = (6x + 7)^4 and evaluate f′(1).
52728
By the chain rule, f′(x) = 4 × 6(6x + 7)^3 = 24(6x + 7)^3. At x = 1, f′(1) = 24 × 13^3 = 52728.
4. Using the chain rule, differentiate f(x) = (2x + 2)^2 and evaluate f′(1).
16
By the chain rule, f′(x) = 2 × 2(2x + 2)^1 = 4(2x + 2)^1. At x = 1, f′(1) = 4 × 4^1 = 16.
5. Using the chain rule, differentiate f(x) = (4x + 4)^2 and evaluate f′(2).
96
By the chain rule, f′(x) = 2 × 4(4x + 4)^1 = 8(4x + 4)^1. At x = 2, f′(2) = 8 × 12^1 = 96.
6. Using the chain rule, differentiate f(x) = (6x + 6)^4 and evaluate f′(3).
331776
By the chain rule, f′(x) = 4 × 6(6x + 6)^3 = 24(6x + 6)^3. At x = 3, f′(3) = 24 × 24^3 = 331776.
7. Using the chain rule, differentiate f(x) = (3x + 4)^4 and evaluate f′(1).
4116
By the chain rule, f′(x) = 4 × 3(3x + 4)^3 = 12(3x + 4)^3. At x = 1, f′(1) = 12 × 7^3 = 4116.
8. Using the chain rule, differentiate f(x) = (4x + 4)^3 and evaluate f′(4).
4800
By the chain rule, f′(x) = 3 × 4(4x + 4)^2 = 12(4x + 4)^2. At x = 4, f′(4) = 12 × 20^2 = 4800.
9. Using the chain rule, differentiate f(x) = (6x + 3)^3 and evaluate f′(1).
1458
By the chain rule, f′(x) = 3 × 6(6x + 3)^2 = 18(6x + 3)^2. At x = 1, f′(1) = 18 × 9^2 = 1458.
10. Using the chain rule, differentiate f(x) = (2x + 1)^3 and evaluate f′(3).
294
By the chain rule, f′(x) = 3 × 2(2x + 1)^2 = 6(2x + 1)^2. At x = 3, f′(3) = 6 × 7^2 = 294.
11. Using the chain rule, differentiate f(x) = (2x + 3)^2 and evaluate f′(3).
36
By the chain rule, f′(x) = 2 × 2(2x + 3)^1 = 4(2x + 3)^1. At x = 3, f′(3) = 4 × 9^1 = 36.
12. Using the chain rule, differentiate f(x) = (4x + 3)^4 and evaluate f′(2).
21296
By the chain rule, f′(x) = 4 × 4(4x + 3)^3 = 16(4x + 3)^3. At x = 2, f′(2) = 16 × 11^3 = 21296.

12 questions · Seed 47 · Age guidance, not curriculum certification