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 + 2)^3 and evaluate f′(4).
f(x) = (6x + 2)^3 · f′(4) at u = 26
2. Using the chain rule, differentiate f(x) = (4x + 2)^3 and evaluate f′(1).
f(x) = (4x + 2)^3 · f′(1) at u = 6
3. Using the chain rule, differentiate f(x) = (3x + 3)^4 and evaluate f′(3).
f(x) = (3x + 3)^4 · f′(3) at u = 12
4. Using the chain rule, differentiate f(x) = (6x + 6)^3 and evaluate f′(1).
f(x) = (6x + 6)^3 · f′(1) at u = 12
5. Using the chain rule, differentiate f(x) = (4x + 7)^4 and evaluate f′(4).
f(x) = (4x + 7)^4 · f′(4) at u = 23
6. Using the chain rule, differentiate f(x) = (5x + 3)^3 and evaluate f′(5).
f(x) = (5x + 3)^3 · f′(5) at u = 28
7. Using the chain rule, differentiate f(x) = (3x + 8)^4 and evaluate f′(1).
f(x) = (3x + 8)^4 · f′(1) at u = 11
8. Using the chain rule, differentiate f(x) = (4x + 4)^3 and evaluate f′(5).
f(x) = (4x + 4)^3 · f′(5) at u = 24
9. Using the chain rule, differentiate f(x) = (6x + 4)^2 and evaluate f′(3).
f(x) = (6x + 4)^2 · f′(3) at u = 22
10. Using the chain rule, differentiate f(x) = (4x + 2)^4 and evaluate f′(1).
f(x) = (4x + 2)^4 · f′(1) at u = 6
11. Using the chain rule, differentiate f(x) = (2x + 3)^2 and evaluate f′(5).
f(x) = (2x + 3)^2 · f′(5) at u = 13
12. Using the chain rule, differentiate f(x) = (3x + 7)^3 and evaluate f′(2).
f(x) = (3x + 7)^3 · f′(2) at u = 13

12 questions · Seed 44 · 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 + 2)^3 and evaluate f′(4).
12168
By the chain rule, f′(x) = 3 × 6(6x + 2)^2 = 18(6x + 2)^2. At x = 4, f′(4) = 18 × 26^2 = 12168.
2. Using the chain rule, differentiate f(x) = (4x + 2)^3 and evaluate f′(1).
432
By the chain rule, f′(x) = 3 × 4(4x + 2)^2 = 12(4x + 2)^2. At x = 1, f′(1) = 12 × 6^2 = 432.
3. Using the chain rule, differentiate f(x) = (3x + 3)^4 and evaluate f′(3).
20736
By the chain rule, f′(x) = 4 × 3(3x + 3)^3 = 12(3x + 3)^3. At x = 3, f′(3) = 12 × 12^3 = 20736.
4. Using the chain rule, differentiate f(x) = (6x + 6)^3 and evaluate f′(1).
2592
By the chain rule, f′(x) = 3 × 6(6x + 6)^2 = 18(6x + 6)^2. At x = 1, f′(1) = 18 × 12^2 = 2592.
5. Using the chain rule, differentiate f(x) = (4x + 7)^4 and evaluate f′(4).
194672
By the chain rule, f′(x) = 4 × 4(4x + 7)^3 = 16(4x + 7)^3. At x = 4, f′(4) = 16 × 23^3 = 194672.
6. Using the chain rule, differentiate f(x) = (5x + 3)^3 and evaluate f′(5).
11760
By the chain rule, f′(x) = 3 × 5(5x + 3)^2 = 15(5x + 3)^2. At x = 5, f′(5) = 15 × 28^2 = 11760.
7. Using the chain rule, differentiate f(x) = (3x + 8)^4 and evaluate f′(1).
15972
By the chain rule, f′(x) = 4 × 3(3x + 8)^3 = 12(3x + 8)^3. At x = 1, f′(1) = 12 × 11^3 = 15972.
8. Using the chain rule, differentiate f(x) = (4x + 4)^3 and evaluate f′(5).
6912
By the chain rule, f′(x) = 3 × 4(4x + 4)^2 = 12(4x + 4)^2. At x = 5, f′(5) = 12 × 24^2 = 6912.
9. Using the chain rule, differentiate f(x) = (6x + 4)^2 and evaluate f′(3).
264
By the chain rule, f′(x) = 2 × 6(6x + 4)^1 = 12(6x + 4)^1. At x = 3, f′(3) = 12 × 22^1 = 264.
10. Using the chain rule, differentiate f(x) = (4x + 2)^4 and evaluate f′(1).
3456
By the chain rule, f′(x) = 4 × 4(4x + 2)^3 = 16(4x + 2)^3. At x = 1, f′(1) = 16 × 6^3 = 3456.
11. Using the chain rule, differentiate f(x) = (2x + 3)^2 and evaluate f′(5).
52
By the chain rule, f′(x) = 2 × 2(2x + 3)^1 = 4(2x + 3)^1. At x = 5, f′(5) = 4 × 13^1 = 52.
12. Using the chain rule, differentiate f(x) = (3x + 7)^3 and evaluate f′(2).
1521
By the chain rule, f′(x) = 3 × 3(3x + 7)^2 = 9(3x + 7)^2. At x = 2, f′(2) = 9 × 13^2 = 1521.

12 questions · Seed 44 · 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 + 2)^3 and evaluate f′(4).
f(x) = (6x + 2)^3 · f′(4) at u = 26
2. Using the chain rule, differentiate f(x) = (4x + 2)^3 and evaluate f′(1).
f(x) = (4x + 2)^3 · f′(1) at u = 6
3. Using the chain rule, differentiate f(x) = (3x + 3)^4 and evaluate f′(3).
f(x) = (3x + 3)^4 · f′(3) at u = 12
4. Using the chain rule, differentiate f(x) = (6x + 6)^3 and evaluate f′(1).
f(x) = (6x + 6)^3 · f′(1) at u = 12
5. Using the chain rule, differentiate f(x) = (4x + 7)^4 and evaluate f′(4).
f(x) = (4x + 7)^4 · f′(4) at u = 23
6. Using the chain rule, differentiate f(x) = (5x + 3)^3 and evaluate f′(5).
f(x) = (5x + 3)^3 · f′(5) at u = 28
7. Using the chain rule, differentiate f(x) = (3x + 8)^4 and evaluate f′(1).
f(x) = (3x + 8)^4 · f′(1) at u = 11
8. Using the chain rule, differentiate f(x) = (4x + 4)^3 and evaluate f′(5).
f(x) = (4x + 4)^3 · f′(5) at u = 24
9. Using the chain rule, differentiate f(x) = (6x + 4)^2 and evaluate f′(3).
f(x) = (6x + 4)^2 · f′(3) at u = 22
10. Using the chain rule, differentiate f(x) = (4x + 2)^4 and evaluate f′(1).
f(x) = (4x + 2)^4 · f′(1) at u = 6
11. Using the chain rule, differentiate f(x) = (2x + 3)^2 and evaluate f′(5).
f(x) = (2x + 3)^2 · f′(5) at u = 13
12. Using the chain rule, differentiate f(x) = (3x + 7)^3 and evaluate f′(2).
f(x) = (3x + 7)^3 · f′(2) at u = 13

12 questions · Seed 44 · 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 + 2)^3 and evaluate f′(4).
12168
By the chain rule, f′(x) = 3 × 6(6x + 2)^2 = 18(6x + 2)^2. At x = 4, f′(4) = 18 × 26^2 = 12168.
2. Using the chain rule, differentiate f(x) = (4x + 2)^3 and evaluate f′(1).
432
By the chain rule, f′(x) = 3 × 4(4x + 2)^2 = 12(4x + 2)^2. At x = 1, f′(1) = 12 × 6^2 = 432.
3. Using the chain rule, differentiate f(x) = (3x + 3)^4 and evaluate f′(3).
20736
By the chain rule, f′(x) = 4 × 3(3x + 3)^3 = 12(3x + 3)^3. At x = 3, f′(3) = 12 × 12^3 = 20736.
4. Using the chain rule, differentiate f(x) = (6x + 6)^3 and evaluate f′(1).
2592
By the chain rule, f′(x) = 3 × 6(6x + 6)^2 = 18(6x + 6)^2. At x = 1, f′(1) = 18 × 12^2 = 2592.
5. Using the chain rule, differentiate f(x) = (4x + 7)^4 and evaluate f′(4).
194672
By the chain rule, f′(x) = 4 × 4(4x + 7)^3 = 16(4x + 7)^3. At x = 4, f′(4) = 16 × 23^3 = 194672.
6. Using the chain rule, differentiate f(x) = (5x + 3)^3 and evaluate f′(5).
11760
By the chain rule, f′(x) = 3 × 5(5x + 3)^2 = 15(5x + 3)^2. At x = 5, f′(5) = 15 × 28^2 = 11760.
7. Using the chain rule, differentiate f(x) = (3x + 8)^4 and evaluate f′(1).
15972
By the chain rule, f′(x) = 4 × 3(3x + 8)^3 = 12(3x + 8)^3. At x = 1, f′(1) = 12 × 11^3 = 15972.
8. Using the chain rule, differentiate f(x) = (4x + 4)^3 and evaluate f′(5).
6912
By the chain rule, f′(x) = 3 × 4(4x + 4)^2 = 12(4x + 4)^2. At x = 5, f′(5) = 12 × 24^2 = 6912.
9. Using the chain rule, differentiate f(x) = (6x + 4)^2 and evaluate f′(3).
264
By the chain rule, f′(x) = 2 × 6(6x + 4)^1 = 12(6x + 4)^1. At x = 3, f′(3) = 12 × 22^1 = 264.
10. Using the chain rule, differentiate f(x) = (4x + 2)^4 and evaluate f′(1).
3456
By the chain rule, f′(x) = 4 × 4(4x + 2)^3 = 16(4x + 2)^3. At x = 1, f′(1) = 16 × 6^3 = 3456.
11. Using the chain rule, differentiate f(x) = (2x + 3)^2 and evaluate f′(5).
52
By the chain rule, f′(x) = 2 × 2(2x + 3)^1 = 4(2x + 3)^1. At x = 5, f′(5) = 4 × 13^1 = 52.
12. Using the chain rule, differentiate f(x) = (3x + 7)^3 and evaluate f′(2).
1521
By the chain rule, f′(x) = 3 × 3(3x + 7)^2 = 9(3x + 7)^2. At x = 2, f′(2) = 9 × 13^2 = 1521.

12 questions · Seed 44 · Age guidance, not curriculum certification