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) = (4x + 3)^4 and evaluate f′(5).
f(x) = (4x + 3)^4 · f′(5) at u = 23
2. Using the chain rule, differentiate f(x) = (3x + 7)^2 and evaluate f′(1).
f(x) = (3x + 7)^2 · f′(1) at u = 10
3. Using the chain rule, differentiate f(x) = (4x + 7)^4 and evaluate f′(2).
f(x) = (4x + 7)^4 · f′(2) at u = 15
4. Using the chain rule, differentiate f(x) = (3x + 1)^2 and evaluate f′(3).
f(x) = (3x + 1)^2 · f′(3) at u = 10
5. Using the chain rule, differentiate f(x) = (3x + 7)^2 and evaluate f′(4).
f(x) = (3x + 7)^2 · f′(4) at u = 19
6. Using the chain rule, differentiate f(x) = (6x + 2)^2 and evaluate f′(1).
f(x) = (6x + 2)^2 · f′(1) at u = 8
7. Using the chain rule, differentiate f(x) = (3x + 1)^2 and evaluate f′(5).
f(x) = (3x + 1)^2 · f′(5) at u = 16
8. Using the chain rule, differentiate f(x) = (6x + 1)^4 and evaluate f′(5).
f(x) = (6x + 1)^4 · f′(5) at u = 31
9. Using the chain rule, differentiate f(x) = (4x + 7)^2 and evaluate f′(1).
f(x) = (4x + 7)^2 · f′(1) at u = 11
10. Using the chain rule, differentiate f(x) = (5x + 1)^3 and evaluate f′(5).
f(x) = (5x + 1)^3 · f′(5) at u = 26
11. Using the chain rule, differentiate f(x) = (3x + 2)^3 and evaluate f′(2).
f(x) = (3x + 2)^3 · f′(2) at u = 8
12. Using the chain rule, differentiate f(x) = (4x + 8)^2 and evaluate f′(4).
f(x) = (4x + 8)^2 · f′(4) at u = 24

12 questions · Seed 43 · 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) = (4x + 3)^4 and evaluate f′(5).
194672
By the chain rule, f′(x) = 4 × 4(4x + 3)^3 = 16(4x + 3)^3. At x = 5, f′(5) = 16 × 23^3 = 194672.
2. Using the chain rule, differentiate f(x) = (3x + 7)^2 and evaluate f′(1).
60
By the chain rule, f′(x) = 2 × 3(3x + 7)^1 = 6(3x + 7)^1. At x = 1, f′(1) = 6 × 10^1 = 60.
3. Using the chain rule, differentiate f(x) = (4x + 7)^4 and evaluate f′(2).
54000
By the chain rule, f′(x) = 4 × 4(4x + 7)^3 = 16(4x + 7)^3. At x = 2, f′(2) = 16 × 15^3 = 54000.
4. Using the chain rule, differentiate f(x) = (3x + 1)^2 and evaluate f′(3).
60
By the chain rule, f′(x) = 2 × 3(3x + 1)^1 = 6(3x + 1)^1. At x = 3, f′(3) = 6 × 10^1 = 60.
5. Using the chain rule, differentiate f(x) = (3x + 7)^2 and evaluate f′(4).
114
By the chain rule, f′(x) = 2 × 3(3x + 7)^1 = 6(3x + 7)^1. At x = 4, f′(4) = 6 × 19^1 = 114.
6. Using the chain rule, differentiate f(x) = (6x + 2)^2 and evaluate f′(1).
96
By the chain rule, f′(x) = 2 × 6(6x + 2)^1 = 12(6x + 2)^1. At x = 1, f′(1) = 12 × 8^1 = 96.
7. Using the chain rule, differentiate f(x) = (3x + 1)^2 and evaluate f′(5).
96
By the chain rule, f′(x) = 2 × 3(3x + 1)^1 = 6(3x + 1)^1. At x = 5, f′(5) = 6 × 16^1 = 96.
8. Using the chain rule, differentiate f(x) = (6x + 1)^4 and evaluate f′(5).
714984
By the chain rule, f′(x) = 4 × 6(6x + 1)^3 = 24(6x + 1)^3. At x = 5, f′(5) = 24 × 31^3 = 714984.
9. Using the chain rule, differentiate f(x) = (4x + 7)^2 and evaluate f′(1).
88
By the chain rule, f′(x) = 2 × 4(4x + 7)^1 = 8(4x + 7)^1. At x = 1, f′(1) = 8 × 11^1 = 88.
10. Using the chain rule, differentiate f(x) = (5x + 1)^3 and evaluate f′(5).
10140
By the chain rule, f′(x) = 3 × 5(5x + 1)^2 = 15(5x + 1)^2. At x = 5, f′(5) = 15 × 26^2 = 10140.
11. Using the chain rule, differentiate f(x) = (3x + 2)^3 and evaluate f′(2).
576
By the chain rule, f′(x) = 3 × 3(3x + 2)^2 = 9(3x + 2)^2. At x = 2, f′(2) = 9 × 8^2 = 576.
12. Using the chain rule, differentiate f(x) = (4x + 8)^2 and evaluate f′(4).
192
By the chain rule, f′(x) = 2 × 4(4x + 8)^1 = 8(4x + 8)^1. At x = 4, f′(4) = 8 × 24^1 = 192.

12 questions · Seed 43 · 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) = (4x + 3)^4 and evaluate f′(5).
f(x) = (4x + 3)^4 · f′(5) at u = 23
2. Using the chain rule, differentiate f(x) = (3x + 7)^2 and evaluate f′(1).
f(x) = (3x + 7)^2 · f′(1) at u = 10
3. Using the chain rule, differentiate f(x) = (4x + 7)^4 and evaluate f′(2).
f(x) = (4x + 7)^4 · f′(2) at u = 15
4. Using the chain rule, differentiate f(x) = (3x + 1)^2 and evaluate f′(3).
f(x) = (3x + 1)^2 · f′(3) at u = 10
5. Using the chain rule, differentiate f(x) = (3x + 7)^2 and evaluate f′(4).
f(x) = (3x + 7)^2 · f′(4) at u = 19
6. Using the chain rule, differentiate f(x) = (6x + 2)^2 and evaluate f′(1).
f(x) = (6x + 2)^2 · f′(1) at u = 8
7. Using the chain rule, differentiate f(x) = (3x + 1)^2 and evaluate f′(5).
f(x) = (3x + 1)^2 · f′(5) at u = 16
8. Using the chain rule, differentiate f(x) = (6x + 1)^4 and evaluate f′(5).
f(x) = (6x + 1)^4 · f′(5) at u = 31
9. Using the chain rule, differentiate f(x) = (4x + 7)^2 and evaluate f′(1).
f(x) = (4x + 7)^2 · f′(1) at u = 11
10. Using the chain rule, differentiate f(x) = (5x + 1)^3 and evaluate f′(5).
f(x) = (5x + 1)^3 · f′(5) at u = 26
11. Using the chain rule, differentiate f(x) = (3x + 2)^3 and evaluate f′(2).
f(x) = (3x + 2)^3 · f′(2) at u = 8
12. Using the chain rule, differentiate f(x) = (4x + 8)^2 and evaluate f′(4).
f(x) = (4x + 8)^2 · f′(4) at u = 24

12 questions · Seed 43 · 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) = (4x + 3)^4 and evaluate f′(5).
194672
By the chain rule, f′(x) = 4 × 4(4x + 3)^3 = 16(4x + 3)^3. At x = 5, f′(5) = 16 × 23^3 = 194672.
2. Using the chain rule, differentiate f(x) = (3x + 7)^2 and evaluate f′(1).
60
By the chain rule, f′(x) = 2 × 3(3x + 7)^1 = 6(3x + 7)^1. At x = 1, f′(1) = 6 × 10^1 = 60.
3. Using the chain rule, differentiate f(x) = (4x + 7)^4 and evaluate f′(2).
54000
By the chain rule, f′(x) = 4 × 4(4x + 7)^3 = 16(4x + 7)^3. At x = 2, f′(2) = 16 × 15^3 = 54000.
4. Using the chain rule, differentiate f(x) = (3x + 1)^2 and evaluate f′(3).
60
By the chain rule, f′(x) = 2 × 3(3x + 1)^1 = 6(3x + 1)^1. At x = 3, f′(3) = 6 × 10^1 = 60.
5. Using the chain rule, differentiate f(x) = (3x + 7)^2 and evaluate f′(4).
114
By the chain rule, f′(x) = 2 × 3(3x + 7)^1 = 6(3x + 7)^1. At x = 4, f′(4) = 6 × 19^1 = 114.
6. Using the chain rule, differentiate f(x) = (6x + 2)^2 and evaluate f′(1).
96
By the chain rule, f′(x) = 2 × 6(6x + 2)^1 = 12(6x + 2)^1. At x = 1, f′(1) = 12 × 8^1 = 96.
7. Using the chain rule, differentiate f(x) = (3x + 1)^2 and evaluate f′(5).
96
By the chain rule, f′(x) = 2 × 3(3x + 1)^1 = 6(3x + 1)^1. At x = 5, f′(5) = 6 × 16^1 = 96.
8. Using the chain rule, differentiate f(x) = (6x + 1)^4 and evaluate f′(5).
714984
By the chain rule, f′(x) = 4 × 6(6x + 1)^3 = 24(6x + 1)^3. At x = 5, f′(5) = 24 × 31^3 = 714984.
9. Using the chain rule, differentiate f(x) = (4x + 7)^2 and evaluate f′(1).
88
By the chain rule, f′(x) = 2 × 4(4x + 7)^1 = 8(4x + 7)^1. At x = 1, f′(1) = 8 × 11^1 = 88.
10. Using the chain rule, differentiate f(x) = (5x + 1)^3 and evaluate f′(5).
10140
By the chain rule, f′(x) = 3 × 5(5x + 1)^2 = 15(5x + 1)^2. At x = 5, f′(5) = 15 × 26^2 = 10140.
11. Using the chain rule, differentiate f(x) = (3x + 2)^3 and evaluate f′(2).
576
By the chain rule, f′(x) = 3 × 3(3x + 2)^2 = 9(3x + 2)^2. At x = 2, f′(2) = 9 × 8^2 = 576.
12. Using the chain rule, differentiate f(x) = (4x + 8)^2 and evaluate f′(4).
192
By the chain rule, f′(x) = 2 × 4(4x + 8)^1 = 8(4x + 8)^1. At x = 4, f′(4) = 8 × 24^1 = 192.

12 questions · Seed 43 · Age guidance, not curriculum certification