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) = (2x + 1)^3 and evaluate f′(3).
f(x) = (2x + 1)^3 · f′(3) at u = 7
2. Using the chain rule, differentiate f(x) = (4x + 6)^3 and evaluate f′(2).
f(x) = (4x + 6)^3 · f′(2) at u = 14
3. Using the chain rule, differentiate f(x) = (4x + 5)^4 and evaluate f′(3).
f(x) = (4x + 5)^4 · f′(3) at u = 17
4. Using the chain rule, differentiate f(x) = (4x + 8)^4 and evaluate f′(1).
f(x) = (4x + 8)^4 · f′(1) at u = 12
5. Using the chain rule, differentiate f(x) = (4x + 1)^2 and evaluate f′(3).
f(x) = (4x + 1)^2 · f′(3) at u = 13
6. Using the chain rule, differentiate f(x) = (6x + 4)^2 and evaluate f′(1).
f(x) = (6x + 4)^2 · f′(1) at u = 10
7. Using the chain rule, differentiate f(x) = (5x + 1)^4 and evaluate f′(5).
f(x) = (5x + 1)^4 · f′(5) at u = 26
8. Using the chain rule, differentiate f(x) = (5x + 8)^4 and evaluate f′(3).
f(x) = (5x + 8)^4 · f′(3) at u = 23
9. Using the chain rule, differentiate f(x) = (6x + 8)^3 and evaluate f′(5).
f(x) = (6x + 8)^3 · f′(5) at u = 38
10. Using the chain rule, differentiate f(x) = (4x + 6)^2 and evaluate f′(2).
f(x) = (4x + 6)^2 · f′(2) at u = 14
11. Using the chain rule, differentiate f(x) = (4x + 6)^2 and evaluate f′(4).
f(x) = (4x + 6)^2 · f′(4) at u = 22
12. Using the chain rule, differentiate f(x) = (5x + 6)^3 and evaluate f′(5).
f(x) = (5x + 6)^3 · f′(5) at u = 31

12 questions · Seed 46 · 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) = (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.
2. Using the chain rule, differentiate f(x) = (4x + 6)^3 and evaluate f′(2).
2352
By the chain rule, f′(x) = 3 × 4(4x + 6)^2 = 12(4x + 6)^2. At x = 2, f′(2) = 12 × 14^2 = 2352.
3. Using the chain rule, differentiate f(x) = (4x + 5)^4 and evaluate f′(3).
78608
By the chain rule, f′(x) = 4 × 4(4x + 5)^3 = 16(4x + 5)^3. At x = 3, f′(3) = 16 × 17^3 = 78608.
4. Using the chain rule, differentiate f(x) = (4x + 8)^4 and evaluate f′(1).
27648
By the chain rule, f′(x) = 4 × 4(4x + 8)^3 = 16(4x + 8)^3. At x = 1, f′(1) = 16 × 12^3 = 27648.
5. Using the chain rule, differentiate f(x) = (4x + 1)^2 and evaluate f′(3).
104
By the chain rule, f′(x) = 2 × 4(4x + 1)^1 = 8(4x + 1)^1. At x = 3, f′(3) = 8 × 13^1 = 104.
6. Using the chain rule, differentiate f(x) = (6x + 4)^2 and evaluate f′(1).
120
By the chain rule, f′(x) = 2 × 6(6x + 4)^1 = 12(6x + 4)^1. At x = 1, f′(1) = 12 × 10^1 = 120.
7. Using the chain rule, differentiate f(x) = (5x + 1)^4 and evaluate f′(5).
351520
By the chain rule, f′(x) = 4 × 5(5x + 1)^3 = 20(5x + 1)^3. At x = 5, f′(5) = 20 × 26^3 = 351520.
8. Using the chain rule, differentiate f(x) = (5x + 8)^4 and evaluate f′(3).
243340
By the chain rule, f′(x) = 4 × 5(5x + 8)^3 = 20(5x + 8)^3. At x = 3, f′(3) = 20 × 23^3 = 243340.
9. Using the chain rule, differentiate f(x) = (6x + 8)^3 and evaluate f′(5).
25992
By the chain rule, f′(x) = 3 × 6(6x + 8)^2 = 18(6x + 8)^2. At x = 5, f′(5) = 18 × 38^2 = 25992.
10. Using the chain rule, differentiate f(x) = (4x + 6)^2 and evaluate f′(2).
112
By the chain rule, f′(x) = 2 × 4(4x + 6)^1 = 8(4x + 6)^1. At x = 2, f′(2) = 8 × 14^1 = 112.
11. Using the chain rule, differentiate f(x) = (4x + 6)^2 and evaluate f′(4).
176
By the chain rule, f′(x) = 2 × 4(4x + 6)^1 = 8(4x + 6)^1. At x = 4, f′(4) = 8 × 22^1 = 176.
12. Using the chain rule, differentiate f(x) = (5x + 6)^3 and evaluate f′(5).
14415
By the chain rule, f′(x) = 3 × 5(5x + 6)^2 = 15(5x + 6)^2. At x = 5, f′(5) = 15 × 31^2 = 14415.

12 questions · Seed 46 · 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) = (2x + 1)^3 and evaluate f′(3).
f(x) = (2x + 1)^3 · f′(3) at u = 7
2. Using the chain rule, differentiate f(x) = (4x + 6)^3 and evaluate f′(2).
f(x) = (4x + 6)^3 · f′(2) at u = 14
3. Using the chain rule, differentiate f(x) = (4x + 5)^4 and evaluate f′(3).
f(x) = (4x + 5)^4 · f′(3) at u = 17
4. Using the chain rule, differentiate f(x) = (4x + 8)^4 and evaluate f′(1).
f(x) = (4x + 8)^4 · f′(1) at u = 12
5. Using the chain rule, differentiate f(x) = (4x + 1)^2 and evaluate f′(3).
f(x) = (4x + 1)^2 · f′(3) at u = 13
6. Using the chain rule, differentiate f(x) = (6x + 4)^2 and evaluate f′(1).
f(x) = (6x + 4)^2 · f′(1) at u = 10
7. Using the chain rule, differentiate f(x) = (5x + 1)^4 and evaluate f′(5).
f(x) = (5x + 1)^4 · f′(5) at u = 26
8. Using the chain rule, differentiate f(x) = (5x + 8)^4 and evaluate f′(3).
f(x) = (5x + 8)^4 · f′(3) at u = 23
9. Using the chain rule, differentiate f(x) = (6x + 8)^3 and evaluate f′(5).
f(x) = (6x + 8)^3 · f′(5) at u = 38
10. Using the chain rule, differentiate f(x) = (4x + 6)^2 and evaluate f′(2).
f(x) = (4x + 6)^2 · f′(2) at u = 14
11. Using the chain rule, differentiate f(x) = (4x + 6)^2 and evaluate f′(4).
f(x) = (4x + 6)^2 · f′(4) at u = 22
12. Using the chain rule, differentiate f(x) = (5x + 6)^3 and evaluate f′(5).
f(x) = (5x + 6)^3 · f′(5) at u = 31

12 questions · Seed 46 · 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) = (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.
2. Using the chain rule, differentiate f(x) = (4x + 6)^3 and evaluate f′(2).
2352
By the chain rule, f′(x) = 3 × 4(4x + 6)^2 = 12(4x + 6)^2. At x = 2, f′(2) = 12 × 14^2 = 2352.
3. Using the chain rule, differentiate f(x) = (4x + 5)^4 and evaluate f′(3).
78608
By the chain rule, f′(x) = 4 × 4(4x + 5)^3 = 16(4x + 5)^3. At x = 3, f′(3) = 16 × 17^3 = 78608.
4. Using the chain rule, differentiate f(x) = (4x + 8)^4 and evaluate f′(1).
27648
By the chain rule, f′(x) = 4 × 4(4x + 8)^3 = 16(4x + 8)^3. At x = 1, f′(1) = 16 × 12^3 = 27648.
5. Using the chain rule, differentiate f(x) = (4x + 1)^2 and evaluate f′(3).
104
By the chain rule, f′(x) = 2 × 4(4x + 1)^1 = 8(4x + 1)^1. At x = 3, f′(3) = 8 × 13^1 = 104.
6. Using the chain rule, differentiate f(x) = (6x + 4)^2 and evaluate f′(1).
120
By the chain rule, f′(x) = 2 × 6(6x + 4)^1 = 12(6x + 4)^1. At x = 1, f′(1) = 12 × 10^1 = 120.
7. Using the chain rule, differentiate f(x) = (5x + 1)^4 and evaluate f′(5).
351520
By the chain rule, f′(x) = 4 × 5(5x + 1)^3 = 20(5x + 1)^3. At x = 5, f′(5) = 20 × 26^3 = 351520.
8. Using the chain rule, differentiate f(x) = (5x + 8)^4 and evaluate f′(3).
243340
By the chain rule, f′(x) = 4 × 5(5x + 8)^3 = 20(5x + 8)^3. At x = 3, f′(3) = 20 × 23^3 = 243340.
9. Using the chain rule, differentiate f(x) = (6x + 8)^3 and evaluate f′(5).
25992
By the chain rule, f′(x) = 3 × 6(6x + 8)^2 = 18(6x + 8)^2. At x = 5, f′(5) = 18 × 38^2 = 25992.
10. Using the chain rule, differentiate f(x) = (4x + 6)^2 and evaluate f′(2).
112
By the chain rule, f′(x) = 2 × 4(4x + 6)^1 = 8(4x + 6)^1. At x = 2, f′(2) = 8 × 14^1 = 112.
11. Using the chain rule, differentiate f(x) = (4x + 6)^2 and evaluate f′(4).
176
By the chain rule, f′(x) = 2 × 4(4x + 6)^1 = 8(4x + 6)^1. At x = 4, f′(4) = 8 × 22^1 = 176.
12. Using the chain rule, differentiate f(x) = (5x + 6)^3 and evaluate f′(5).
14415
By the chain rule, f′(x) = 3 × 5(5x + 6)^2 = 15(5x + 6)^2. At x = 5, f′(5) = 15 × 31^2 = 14415.

12 questions · Seed 46 · Age guidance, not curriculum certification