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 + 7)^2 and evaluate f′(3).
f(x) = (4x + 7)^2 · f′(3) at u = 19
2. Using the chain rule, differentiate f(x) = (3x + 1)^2 and evaluate f′(1).
f(x) = (3x + 1)^2 · f′(1) at u = 4
3. Using the chain rule, differentiate f(x) = (4x + 2)^2 and evaluate f′(1).
f(x) = (4x + 2)^2 · f′(1) at u = 6
4. Using the chain rule, differentiate f(x) = (3x + 4)^2 and evaluate f′(2).
f(x) = (3x + 4)^2 · f′(2) at u = 10
5. Using the chain rule, differentiate f(x) = (4x + 3)^3 and evaluate f′(3).
f(x) = (4x + 3)^3 · f′(3) at u = 15
6. Using the chain rule, differentiate f(x) = (5x + 3)^4 and evaluate f′(4).
f(x) = (5x + 3)^4 · f′(4) at u = 23
7. Using the chain rule, differentiate f(x) = (3x + 8)^2 and evaluate f′(3).
f(x) = (3x + 8)^2 · f′(3) at u = 17
8. Using the chain rule, differentiate f(x) = (4x + 7)^2 and evaluate f′(1).
f(x) = (4x + 7)^2 · f′(1) at u = 11
9. Using the chain rule, differentiate f(x) = (6x + 8)^2 and evaluate f′(1).
f(x) = (6x + 8)^2 · f′(1) at u = 14
10. Using the chain rule, differentiate f(x) = (4x + 2)^2 and evaluate f′(2).
f(x) = (4x + 2)^2 · f′(2) at u = 10
11. Using the chain rule, differentiate f(x) = (4x + 7)^2 and evaluate f′(4).
f(x) = (4x + 7)^2 · f′(4) at u = 23
12. Using the chain rule, differentiate f(x) = (3x + 5)^2 and evaluate f′(4).
f(x) = (3x + 5)^2 · f′(4) at u = 17

12 questions · Seed 45 · 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 + 7)^2 and evaluate f′(3).
152
By the chain rule, f′(x) = 2 × 4(4x + 7)^1 = 8(4x + 7)^1. At x = 3, f′(3) = 8 × 19^1 = 152.
2. Using the chain rule, differentiate f(x) = (3x + 1)^2 and evaluate f′(1).
24
By the chain rule, f′(x) = 2 × 3(3x + 1)^1 = 6(3x + 1)^1. At x = 1, f′(1) = 6 × 4^1 = 24.
3. Using the chain rule, differentiate f(x) = (4x + 2)^2 and evaluate f′(1).
48
By the chain rule, f′(x) = 2 × 4(4x + 2)^1 = 8(4x + 2)^1. At x = 1, f′(1) = 8 × 6^1 = 48.
4. Using the chain rule, differentiate f(x) = (3x + 4)^2 and evaluate f′(2).
60
By the chain rule, f′(x) = 2 × 3(3x + 4)^1 = 6(3x + 4)^1. At x = 2, f′(2) = 6 × 10^1 = 60.
5. Using the chain rule, differentiate f(x) = (4x + 3)^3 and evaluate f′(3).
2700
By the chain rule, f′(x) = 3 × 4(4x + 3)^2 = 12(4x + 3)^2. At x = 3, f′(3) = 12 × 15^2 = 2700.
6. Using the chain rule, differentiate f(x) = (5x + 3)^4 and evaluate f′(4).
243340
By the chain rule, f′(x) = 4 × 5(5x + 3)^3 = 20(5x + 3)^3. At x = 4, f′(4) = 20 × 23^3 = 243340.
7. Using the chain rule, differentiate f(x) = (3x + 8)^2 and evaluate f′(3).
102
By the chain rule, f′(x) = 2 × 3(3x + 8)^1 = 6(3x + 8)^1. At x = 3, f′(3) = 6 × 17^1 = 102.
8. 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.
9. Using the chain rule, differentiate f(x) = (6x + 8)^2 and evaluate f′(1).
168
By the chain rule, f′(x) = 2 × 6(6x + 8)^1 = 12(6x + 8)^1. At x = 1, f′(1) = 12 × 14^1 = 168.
10. Using the chain rule, differentiate f(x) = (4x + 2)^2 and evaluate f′(2).
80
By the chain rule, f′(x) = 2 × 4(4x + 2)^1 = 8(4x + 2)^1. At x = 2, f′(2) = 8 × 10^1 = 80.
11. Using the chain rule, differentiate f(x) = (4x + 7)^2 and evaluate f′(4).
184
By the chain rule, f′(x) = 2 × 4(4x + 7)^1 = 8(4x + 7)^1. At x = 4, f′(4) = 8 × 23^1 = 184.
12. Using the chain rule, differentiate f(x) = (3x + 5)^2 and evaluate f′(4).
102
By the chain rule, f′(x) = 2 × 3(3x + 5)^1 = 6(3x + 5)^1. At x = 4, f′(4) = 6 × 17^1 = 102.

12 questions · Seed 45 · 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 + 7)^2 and evaluate f′(3).
f(x) = (4x + 7)^2 · f′(3) at u = 19
2. Using the chain rule, differentiate f(x) = (3x + 1)^2 and evaluate f′(1).
f(x) = (3x + 1)^2 · f′(1) at u = 4
3. Using the chain rule, differentiate f(x) = (4x + 2)^2 and evaluate f′(1).
f(x) = (4x + 2)^2 · f′(1) at u = 6
4. Using the chain rule, differentiate f(x) = (3x + 4)^2 and evaluate f′(2).
f(x) = (3x + 4)^2 · f′(2) at u = 10
5. Using the chain rule, differentiate f(x) = (4x + 3)^3 and evaluate f′(3).
f(x) = (4x + 3)^3 · f′(3) at u = 15
6. Using the chain rule, differentiate f(x) = (5x + 3)^4 and evaluate f′(4).
f(x) = (5x + 3)^4 · f′(4) at u = 23
7. Using the chain rule, differentiate f(x) = (3x + 8)^2 and evaluate f′(3).
f(x) = (3x + 8)^2 · f′(3) at u = 17
8. Using the chain rule, differentiate f(x) = (4x + 7)^2 and evaluate f′(1).
f(x) = (4x + 7)^2 · f′(1) at u = 11
9. Using the chain rule, differentiate f(x) = (6x + 8)^2 and evaluate f′(1).
f(x) = (6x + 8)^2 · f′(1) at u = 14
10. Using the chain rule, differentiate f(x) = (4x + 2)^2 and evaluate f′(2).
f(x) = (4x + 2)^2 · f′(2) at u = 10
11. Using the chain rule, differentiate f(x) = (4x + 7)^2 and evaluate f′(4).
f(x) = (4x + 7)^2 · f′(4) at u = 23
12. Using the chain rule, differentiate f(x) = (3x + 5)^2 and evaluate f′(4).
f(x) = (3x + 5)^2 · f′(4) at u = 17

12 questions · Seed 45 · 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 + 7)^2 and evaluate f′(3).
152
By the chain rule, f′(x) = 2 × 4(4x + 7)^1 = 8(4x + 7)^1. At x = 3, f′(3) = 8 × 19^1 = 152.
2. Using the chain rule, differentiate f(x) = (3x + 1)^2 and evaluate f′(1).
24
By the chain rule, f′(x) = 2 × 3(3x + 1)^1 = 6(3x + 1)^1. At x = 1, f′(1) = 6 × 4^1 = 24.
3. Using the chain rule, differentiate f(x) = (4x + 2)^2 and evaluate f′(1).
48
By the chain rule, f′(x) = 2 × 4(4x + 2)^1 = 8(4x + 2)^1. At x = 1, f′(1) = 8 × 6^1 = 48.
4. Using the chain rule, differentiate f(x) = (3x + 4)^2 and evaluate f′(2).
60
By the chain rule, f′(x) = 2 × 3(3x + 4)^1 = 6(3x + 4)^1. At x = 2, f′(2) = 6 × 10^1 = 60.
5. Using the chain rule, differentiate f(x) = (4x + 3)^3 and evaluate f′(3).
2700
By the chain rule, f′(x) = 3 × 4(4x + 3)^2 = 12(4x + 3)^2. At x = 3, f′(3) = 12 × 15^2 = 2700.
6. Using the chain rule, differentiate f(x) = (5x + 3)^4 and evaluate f′(4).
243340
By the chain rule, f′(x) = 4 × 5(5x + 3)^3 = 20(5x + 3)^3. At x = 4, f′(4) = 20 × 23^3 = 243340.
7. Using the chain rule, differentiate f(x) = (3x + 8)^2 and evaluate f′(3).
102
By the chain rule, f′(x) = 2 × 3(3x + 8)^1 = 6(3x + 8)^1. At x = 3, f′(3) = 6 × 17^1 = 102.
8. 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.
9. Using the chain rule, differentiate f(x) = (6x + 8)^2 and evaluate f′(1).
168
By the chain rule, f′(x) = 2 × 6(6x + 8)^1 = 12(6x + 8)^1. At x = 1, f′(1) = 12 × 14^1 = 168.
10. Using the chain rule, differentiate f(x) = (4x + 2)^2 and evaluate f′(2).
80
By the chain rule, f′(x) = 2 × 4(4x + 2)^1 = 8(4x + 2)^1. At x = 2, f′(2) = 8 × 10^1 = 80.
11. Using the chain rule, differentiate f(x) = (4x + 7)^2 and evaluate f′(4).
184
By the chain rule, f′(x) = 2 × 4(4x + 7)^1 = 8(4x + 7)^1. At x = 4, f′(4) = 8 × 23^1 = 184.
12. Using the chain rule, differentiate f(x) = (3x + 5)^2 and evaluate f′(4).
102
By the chain rule, f′(x) = 2 × 3(3x + 5)^1 = 6(3x + 5)^1. At x = 4, f′(4) = 6 × 17^1 = 102.

12 questions · Seed 45 · Age guidance, not curriculum certification