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) = (5x + 6)^2 and evaluate f′(5).
f(x) = (5x + 6)^2 · f′(5) at u = 31
2. Using the chain rule, differentiate f(x) = (3x + 7)^2 and evaluate f′(3).
f(x) = (3x + 7)^2 · f′(3) at u = 16
3. Using the chain rule, differentiate f(x) = (3x + 6)^4 and evaluate f′(1).
f(x) = (3x + 6)^4 · f′(1) at u = 9
4. Using the chain rule, differentiate f(x) = (6x + 8)^4 and evaluate f′(4).
f(x) = (6x + 8)^4 · f′(4) at u = 32
5. Using the chain rule, differentiate f(x) = (2x + 5)^4 and evaluate f′(5).
f(x) = (2x + 5)^4 · f′(5) at u = 15
6. Using the chain rule, differentiate f(x) = (6x + 2)^4 and evaluate f′(1).
f(x) = (6x + 2)^4 · f′(1) at u = 8
7. Using the chain rule, differentiate f(x) = (4x + 3)^2 and evaluate f′(1).
f(x) = (4x + 3)^2 · f′(1) at u = 7
8. Using the chain rule, differentiate f(x) = (2x + 8)^2 and evaluate f′(3).
f(x) = (2x + 8)^2 · f′(3) at u = 14
9. Using the chain rule, differentiate f(x) = (5x + 5)^3 and evaluate f′(1).
f(x) = (5x + 5)^3 · f′(1) at u = 10
10. Using the chain rule, differentiate f(x) = (2x + 4)^2 and evaluate f′(2).
f(x) = (2x + 4)^2 · f′(2) at u = 8
11. Using the chain rule, differentiate f(x) = (5x + 8)^3 and evaluate f′(2).
f(x) = (5x + 8)^3 · f′(2) at u = 18
12. Using the chain rule, differentiate f(x) = (4x + 6)^4 and evaluate f′(2).
f(x) = (4x + 6)^4 · f′(2) at u = 14

12 questions · Seed 42 · 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) = (5x + 6)^2 and evaluate f′(5).
310
By the chain rule, f′(x) = 2 × 5(5x + 6)^1 = 10(5x + 6)^1. At x = 5, f′(5) = 10 × 31^1 = 310.
2. Using the chain rule, differentiate f(x) = (3x + 7)^2 and evaluate f′(3).
96
By the chain rule, f′(x) = 2 × 3(3x + 7)^1 = 6(3x + 7)^1. At x = 3, f′(3) = 6 × 16^1 = 96.
3. Using the chain rule, differentiate f(x) = (3x + 6)^4 and evaluate f′(1).
8748
By the chain rule, f′(x) = 4 × 3(3x + 6)^3 = 12(3x + 6)^3. At x = 1, f′(1) = 12 × 9^3 = 8748.
4. Using the chain rule, differentiate f(x) = (6x + 8)^4 and evaluate f′(4).
786432
By the chain rule, f′(x) = 4 × 6(6x + 8)^3 = 24(6x + 8)^3. At x = 4, f′(4) = 24 × 32^3 = 786432.
5. Using the chain rule, differentiate f(x) = (2x + 5)^4 and evaluate f′(5).
27000
By the chain rule, f′(x) = 4 × 2(2x + 5)^3 = 8(2x + 5)^3. At x = 5, f′(5) = 8 × 15^3 = 27000.
6. Using the chain rule, differentiate f(x) = (6x + 2)^4 and evaluate f′(1).
12288
By the chain rule, f′(x) = 4 × 6(6x + 2)^3 = 24(6x + 2)^3. At x = 1, f′(1) = 24 × 8^3 = 12288.
7. Using the chain rule, differentiate f(x) = (4x + 3)^2 and evaluate f′(1).
56
By the chain rule, f′(x) = 2 × 4(4x + 3)^1 = 8(4x + 3)^1. At x = 1, f′(1) = 8 × 7^1 = 56.
8. Using the chain rule, differentiate f(x) = (2x + 8)^2 and evaluate f′(3).
56
By the chain rule, f′(x) = 2 × 2(2x + 8)^1 = 4(2x + 8)^1. At x = 3, f′(3) = 4 × 14^1 = 56.
9. Using the chain rule, differentiate f(x) = (5x + 5)^3 and evaluate f′(1).
1500
By the chain rule, f′(x) = 3 × 5(5x + 5)^2 = 15(5x + 5)^2. At x = 1, f′(1) = 15 × 10^2 = 1500.
10. Using the chain rule, differentiate f(x) = (2x + 4)^2 and evaluate f′(2).
32
By the chain rule, f′(x) = 2 × 2(2x + 4)^1 = 4(2x + 4)^1. At x = 2, f′(2) = 4 × 8^1 = 32.
11. Using the chain rule, differentiate f(x) = (5x + 8)^3 and evaluate f′(2).
4860
By the chain rule, f′(x) = 3 × 5(5x + 8)^2 = 15(5x + 8)^2. At x = 2, f′(2) = 15 × 18^2 = 4860.
12. Using the chain rule, differentiate f(x) = (4x + 6)^4 and evaluate f′(2).
43904
By the chain rule, f′(x) = 4 × 4(4x + 6)^3 = 16(4x + 6)^3. At x = 2, f′(2) = 16 × 14^3 = 43904.

12 questions · Seed 42 · 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) = (5x + 6)^2 and evaluate f′(5).
f(x) = (5x + 6)^2 · f′(5) at u = 31
2. Using the chain rule, differentiate f(x) = (3x + 7)^2 and evaluate f′(3).
f(x) = (3x + 7)^2 · f′(3) at u = 16
3. Using the chain rule, differentiate f(x) = (3x + 6)^4 and evaluate f′(1).
f(x) = (3x + 6)^4 · f′(1) at u = 9
4. Using the chain rule, differentiate f(x) = (6x + 8)^4 and evaluate f′(4).
f(x) = (6x + 8)^4 · f′(4) at u = 32
5. Using the chain rule, differentiate f(x) = (2x + 5)^4 and evaluate f′(5).
f(x) = (2x + 5)^4 · f′(5) at u = 15
6. Using the chain rule, differentiate f(x) = (6x + 2)^4 and evaluate f′(1).
f(x) = (6x + 2)^4 · f′(1) at u = 8
7. Using the chain rule, differentiate f(x) = (4x + 3)^2 and evaluate f′(1).
f(x) = (4x + 3)^2 · f′(1) at u = 7
8. Using the chain rule, differentiate f(x) = (2x + 8)^2 and evaluate f′(3).
f(x) = (2x + 8)^2 · f′(3) at u = 14
9. Using the chain rule, differentiate f(x) = (5x + 5)^3 and evaluate f′(1).
f(x) = (5x + 5)^3 · f′(1) at u = 10
10. Using the chain rule, differentiate f(x) = (2x + 4)^2 and evaluate f′(2).
f(x) = (2x + 4)^2 · f′(2) at u = 8
11. Using the chain rule, differentiate f(x) = (5x + 8)^3 and evaluate f′(2).
f(x) = (5x + 8)^3 · f′(2) at u = 18
12. Using the chain rule, differentiate f(x) = (4x + 6)^4 and evaluate f′(2).
f(x) = (4x + 6)^4 · f′(2) at u = 14

12 questions · Seed 42 · 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) = (5x + 6)^2 and evaluate f′(5).
310
By the chain rule, f′(x) = 2 × 5(5x + 6)^1 = 10(5x + 6)^1. At x = 5, f′(5) = 10 × 31^1 = 310.
2. Using the chain rule, differentiate f(x) = (3x + 7)^2 and evaluate f′(3).
96
By the chain rule, f′(x) = 2 × 3(3x + 7)^1 = 6(3x + 7)^1. At x = 3, f′(3) = 6 × 16^1 = 96.
3. Using the chain rule, differentiate f(x) = (3x + 6)^4 and evaluate f′(1).
8748
By the chain rule, f′(x) = 4 × 3(3x + 6)^3 = 12(3x + 6)^3. At x = 1, f′(1) = 12 × 9^3 = 8748.
4. Using the chain rule, differentiate f(x) = (6x + 8)^4 and evaluate f′(4).
786432
By the chain rule, f′(x) = 4 × 6(6x + 8)^3 = 24(6x + 8)^3. At x = 4, f′(4) = 24 × 32^3 = 786432.
5. Using the chain rule, differentiate f(x) = (2x + 5)^4 and evaluate f′(5).
27000
By the chain rule, f′(x) = 4 × 2(2x + 5)^3 = 8(2x + 5)^3. At x = 5, f′(5) = 8 × 15^3 = 27000.
6. Using the chain rule, differentiate f(x) = (6x + 2)^4 and evaluate f′(1).
12288
By the chain rule, f′(x) = 4 × 6(6x + 2)^3 = 24(6x + 2)^3. At x = 1, f′(1) = 24 × 8^3 = 12288.
7. Using the chain rule, differentiate f(x) = (4x + 3)^2 and evaluate f′(1).
56
By the chain rule, f′(x) = 2 × 4(4x + 3)^1 = 8(4x + 3)^1. At x = 1, f′(1) = 8 × 7^1 = 56.
8. Using the chain rule, differentiate f(x) = (2x + 8)^2 and evaluate f′(3).
56
By the chain rule, f′(x) = 2 × 2(2x + 8)^1 = 4(2x + 8)^1. At x = 3, f′(3) = 4 × 14^1 = 56.
9. Using the chain rule, differentiate f(x) = (5x + 5)^3 and evaluate f′(1).
1500
By the chain rule, f′(x) = 3 × 5(5x + 5)^2 = 15(5x + 5)^2. At x = 1, f′(1) = 15 × 10^2 = 1500.
10. Using the chain rule, differentiate f(x) = (2x + 4)^2 and evaluate f′(2).
32
By the chain rule, f′(x) = 2 × 2(2x + 4)^1 = 4(2x + 4)^1. At x = 2, f′(2) = 4 × 8^1 = 32.
11. Using the chain rule, differentiate f(x) = (5x + 8)^3 and evaluate f′(2).
4860
By the chain rule, f′(x) = 3 × 5(5x + 8)^2 = 15(5x + 8)^2. At x = 2, f′(2) = 15 × 18^2 = 4860.
12. Using the chain rule, differentiate f(x) = (4x + 6)^4 and evaluate f′(2).
43904
By the chain rule, f′(x) = 4 × 4(4x + 6)^3 = 16(4x + 6)^3. At x = 2, f′(2) = 16 × 14^3 = 43904.

12 questions · Seed 42 · Age guidance, not curriculum certification