Calculus Worksheets | Differentiation Rules Worksheets - Free Printable

Let’s solve each problem one by one using the Chain Rule.

The Chain Rule says:
If you have a function like y = [something]^n, then its derivative is:
dy/dx = n × [something]^(n-1) × (derivative of “something”)

We’ll apply this to each problem.

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1) y = -(-4x² - 7)^6

Let u = -4x² - 7 → du/dx = -8x
y = -u^6 → dy/du = -6u^5
So dy/dx = dy/du × du/dx = -6u^5 × (-8x) = 48x u^5
Substitute back:
→ 48x(-4x² - 7)^5

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2) y = (1/2)(x² - 5x)^6

Let u = x² - 5x → du/dx = 2x - 5
y = (1/2)u^6 → dy/du = (1/2)*6u^5 = 3u^5
dy/dx = 3u^5 × (2x - 5)
→ 3(x² - 5x)^5(2x - 5)

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3) y = 2(2x³ - 5x)^6

Let u = 2x³ - 5x → du/dx = 6x² - 5
y = 2u^6 → dy/du = 12u^5
dy/dx = 12u^5 × (6x² - 5)
→ 12(2x³ - 5x)^5(6x² - 5)

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4) y = -(5/2)(9x - 4)^3

Let u = 9x - 4 → du/dx = 9
y = -(5/2)u^3 → dy/du = -(5/2)*3u^2 = -(15/2)u^2
dy/dx = -(15/2)u^2 × 9 = -(135/2)u^2
→ -(135/2)(9x - 4)^2

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5) y = -(7x² - 4)^3

Let u = 7x² - 4 → du/dx = 14x
y = -u^3 → dy/du = -3u^2
dy/dx = -3u^2 × 14x = -42x u^2
→ -42x(7x² - 4)^2

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6) y = -6(-4x² - 7x)^4

Let u = -4x² - 7x → du/dx = -8x - 7
y = -6u^4 → dy/du = -24u^3
dy/dx = -24u^3 × (-8x - 7) = 24u^3(8x + 7) [since -(-8x-7)=8x+7]
→ 24(-4x² - 7x)^3(8x + 7)

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7) y = (5/2)(6x³ + 3x)^6

Let u = 6x³ + 3x → du/dx = 18x² + 3
y = (5/2)u^6 → dy/du = (5/2)*6u^5 = 15u^5
dy/dx = 15u^5 × (18x² + 3)
→ 15(6x³ + 3x)^5(18x² + 3)

*(Note: You can factor out 3 from (18x² + 3) if needed, but it’s not required unless specified.)*

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8) y = -(5/2)(4x² - 3)^6

Let u = 4x² - 3 → du/dx = 8x
y = -(5/2)u^6 → dy/du = -(5/2)*6u^5 = -15u^5
dy/dx = -15u^5 × 8x = -120x u^5
→ -120x(4x² - 3)^5

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9) y = 3(-x² - 4)^4

Let u = -x² - 4 → du/dx = -2x
y = 3u^4 → dy/du = 12u^3
dy/dx = 12u^3 × (-2x) = -24x u^3
→ -24x(-x² - 4)^3

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10) y = 3(7x² + 2)^6

Let u = 7x² + 2 → du/dx = 14x
y = 3u^6 → dy/du = 18u^5
dy/dx = 18u^5 × 14x = 252x u^5
→ 252x(7x² + 2)^5

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Final Answer:

1) 48x(-4x² - 7)^5
2) 3(x² - 5x)^5(2x - 5)
3) 12(2x³ - 5x)^5(6x² - 5)
4) -(135/2)(9x - 4)^2
5) -42x(7x² - 4)^2
6) 24(-4x² - 7x)^3(8x + 7)
7) 15(6x³ + 3x)^5(18x² + 3)
8) -120x(4x² - 3)^5
9) -24x(-x² - 4)^3
10) 252x(7x² + 2)^5