Derivative of a Composite Function

Derivative of a Composite Function. Practice questions to deepen understanding of the derivative of a composite function. Online math practice with full solutions and step-by-step explanations.

Derivative of a Composite Function — the chain rule, differentiating nested functions. Comprehensive practice with explanations and examples.

20 questions

Question 1

5.00 pts

Given \(h(x)=(3x+1)^4\). Find \(h\''(x)\).

\(h'(x)=12(3x+1)^3\)

\(h'(x)=4(3x+1)^3\)

\(h'(x)=3(3x+1)^3\)

\(h'(x)=12(3x+1)^4\)

Explanation:

Solution: Chain rule: \(4(3x+1)^3\cdot3=12(3x+1)^3\).

Question 2

5.00 pts

Given \(h(x)=(2x-5)^5\). Find \(h\''(x)\).

\(h'(x)=10(2x-5)^4\)

\(h'(x)=5(2x-5)^4\)

\(h'(x)=10(2x-5)^5\)

\(h'(x)=8(2x-5)^4\)

Explanation:

Solution: Chain rule: \(5(2x-5)^4\cdot2=10(2x-5)^4\).

Question 3

5.00 pts

Given \(h(x)=(x-7)^6\). Find \(h\''(x)\).

\(h'(x)=6(x-7)^5\)

\(h'(x)=6(x-7)^6\)

\(h'(x)=(x-7)^5\)

\(h'(x)=6x^5\)

Explanation:

Solution: Chain rule: \(6(x-7)^5\cdot1=6(x-7)^5\).

Question 4

5.00 pts

Given \(h(x)=(4x+3)^2\). Find \(h\''(x)\).

\(h'(x)=8(4x+3)\)

\(h'(x)=2(4x+3)\)

\(h'(x)=8(4x+3)^2\)

\(h'(x)=4(4x+3)\)

Explanation:

Solution: Chain rule: \(2(4x+3)\cdot4=8(4x+3)\).

Question 5

5.00 pts

Given \(h(x)=(5x-2)^3\). Find \(h\''(x)\).

\(h'(x)=15(5x-2)^2\)

\(h'(x)=3(5x-2)^2\)

\(h'(x)=15(5x-2)^3\)

\(h'(x)=10(5x-2)^2\)

Explanation:

Solution: Chain rule: \(3(5x-2)^2\cdot5=15(5x-2)^2\).

Question 6

5.00 pts

Given \(h(x)=(1-2x)^4\). Find \(h\''(x)\).

\(h'(x)=-8(1-2x)^3\)

\(h'(x)=8(1-2x)^3\)

\(h'(x)=4(1-2x)^3\)

\(h'(x)=-8(1-2x)^4\)

Explanation:

Solution: Chain rule: \(4(1-2x)^3\cdot(-2)=-8(1-2x)^3\).

Question 7

5.00 pts

Given \(h(x)=(x^2+1)^3\). Find \(h\''(x)\).

\(h'(x)=6x(x^2+1)^2\)

\(h'(x)=3(x^2+1)^2\)

\(h'(x)=6x(x^2+1)^3\)

\(h'(x)=3x(x^2+1)^2\)

Explanation:

Solution: Chain rule: \(3(x^2+1)^2\cdot2x=6x(x^2+1)^2\).

Question 8

5.00 pts

Given \(h(x)=(x^2-4x)^2\). Find \(h\''(x)\).

\(h'(x)=2(x^2-4x)(2x-4)\)

\(h'(x)=(2x-4)^2\)

\(h'(x)=4x(x^2-4x)\)

\(h'(x)=2(x-4)\)

Explanation:

Solution: Chain rule: \(2(x^2-4x)\cdot(2x-4)\).

Question 9

5.00 pts

Given \(h(x)=(x^2+3x+2)^4\). Find \(h\''(x)\).

\(h'(x)=4(x^2+3x+2)^3(2x+3)\)

\(h'(x)=4(x^2+3x+2)^3\)

\(h'(x)=8x(x^2+3x+2)^3\)

\(h'(x)=(2x+3)^4\)

Explanation:

Solution: Chain rule: \(4(x^2+3x+2)^3\cdot(2x+3)\).

Question 10

5.00 pts

Given \(h(x)=(2x^2-1)^3\). Find \(h\''(x)\).

\(h'(x)=12x(2x^2-1)^2\)

\(h'(x)=6x(2x^2-1)^2\)

\(h'(x)=3(2x^2-1)^2\)

\(h'(x)=12x(2x^2-1)^3\)

Explanation:

Solution: Chain rule: \(3(2x^2-1)^2\cdot4x=12x(2x^2-1)^2\).

Question 11

5.00 pts

Given \(h(x)=(x^3+1)^2\). Find \(h\''(x)\).

\(h'(x)=6x^2(x^3+1)\)

\(h'(x)=2(x^3+1)\)

\(h'(x)=3x^2(x^3+1)\)

\(h'(x)=6x^2(x^3+1)^2\)

Explanation:

Solution: Chain rule: \(2(x^3+1)\cdot3x^2=6x^2(x^3+1)\).

Question 12

5.00 pts

Given \(h(x)=(x^2+1)^5\). Find \(h\''(x)\).

\(h'(x)=10x(x^2+1)^4\)

\(h'(x)=5(x^2+1)^4\)

\(h'(x)=10x(x^2+1)^5\)

\(h'(x)=2x(x^2+1)^4\)

Explanation:

Solution: Chain rule: \(5(x^2+1)^4\cdot2x=10x(x^2+1)^4\).

Question 13

5.00 pts

Given \(h(x)=(3x+2)^4\). Find \(h\''(1)\).

\(h'(1)=1500\)

\(h'(1)=500\)

\(h'(1)=2000\)

\(h'(1)=1200\)

Explanation:

Solution: \(h\''=12(3x+2)^3\). At x=1: \(12(5)^3=12\cdot125=1500\).

Question 14

5.00 pts

Given \(h(x)=(x^2-1)^3\). Find \(h\''(2)\).

\(h'(2)=108\)

\(h'(2)=96\)

\(h'(2)=72\)

\(h'(2)=120\)

Explanation:

Solution: \(h\''=6x(x^2-1)^2\). At x=2: \(12\cdot9=108\).

Question 15

5.00 pts

Given \(h(x)=(2x-1)^5\). Find \(h\''(0)\).

\(h'(0)=10\)

\(h'(0)=-10\)

\(h'(0)=5\)

\(h'(0)=-5\)

Explanation:

Solution: \(h\''=10(2x-1)^4\). At x=0: \(10(-1)^4=10\).

Question 16

5.00 pts

Given \(h(x)=(x^2+4x+5)^2\). Find \(h\''(-1)\).

\(h'(-1)=8\)

\(h'(-1)=4\)

\(h'(-1)=-8\)

\(h'(-1)=2\)

Explanation:

Solution: \(h\''=2(x^2+4x+5)(2x+4)\). At x=-1: \(2(1-4+5)(2)=2\cdot2\cdot2=8\).

Question 17

5.00 pts

Given \(h(x)=(x^3-2x)^4\). Find \(h\''(x)\).

\(h'(x)=4(x^3-2x)^3(3x^2-2)\)

\(h'(x)=4(x^3-2x)^3\)

\(h'(x)=12x^2(x^3-2x)^3\)

\(h'(x)=(3x^2-2)^4\)

Explanation:

Solution: Chain rule: \(4(x^3-2x)^3\cdot(3x^2-2)\).

Question 18

5.00 pts

Given \(h(x)=(x^2(1+x))^3\). Find \(h\''(x)\).

\(h'(x)=3(x^2(1+x))^2(2x+3x^2)\)

\(h'(x)=3x^2(1+x)^2\)

\(h'(x)=6x(1+x)^3\)

\(h'(x)=3(1+x)^2\cdot2x\)

Explanation:

Solution: Chain rule: inner = x²+x³, inner' = 2x+3x². Result: \(3(x^2+x^3)^2(2x+3x^2)\).

Question 19

5.00 pts

Given \(h(x)=(x^3+3x)^3\). Find \(h\''(x)\).

\(h'(x)=3(x^3+3x)^2(3x^2+3)\)

\(h'(x)=9x^2(x^3+3x)^2\)

\(h'(x)=3(3x^2+3)^3\)

\(h'(x)=(3x^2+3)^3\)

Explanation:

Solution: Chain rule: \(3(x^3+3x)^2\cdot(3x^2+3)\).

Question 20

5.00 pts

Given \(h(x)=(x^4-5x^2+1)^2\). Find \(h\''(x)\).

\(h'(x)=2(x^4-5x^2+1)(4x^3-10x)\)

\(h'(x)=2(4x^3-10x)^2\)

\(h'(x)=(x^4-5x^2+1)(4x^3)\)

\(h'(x)=4x^3-10x\)

Explanation:

Solution: Chain rule: \(2(x^4-5x^2+1)\cdot(4x^3-10x)\).