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Quiz: The Chain Rule

Test your understanding of the Chain Rule with these review questions.

1. The Chain Rule is used to differentiate:

  1. Products of functions
  2. Quotients of functions
  3. Composite functions
  4. Sums of functions
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The correct answer is C. The Chain Rule is specifically for differentiating composite functions—functions of the form f(g(x)) where one function is "inside" another.

Concept Tested: Chain Rule

2. If y = f(g(x)), then dy/dx equals:

  1. f'(x) · g'(x)
  2. f'(g(x)) · g'(x)
  3. f(g'(x))
  4. f'(g(x)) + g'(x)
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The correct answer is B. The Chain Rule formula: d/dx[f(g(x))] = f'(g(x)) · g'(x). "Derivative of outside (evaluated at inside) times derivative of inside."

Concept Tested: Chain Rule Formula

3. What is d/dx[sin(3x)]?

  1. cos(3x)
  2. 3cos(3x)
  3. cos(3)
  4. 3sin(3x)
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The correct answer is B. Using the Chain Rule: outer function is sin, inner is 3x. d/dx[sin(3x)] = cos(3x) · 3 = 3cos(3x).

Concept Tested: Trig Chain Rule

4. What is d/dx[(x² + 1)⁵]?

  1. 5(x² + 1)⁴
  2. 5(2x)⁴
  3. 10x(x² + 1)⁴
  4. 5x(x² + 1)⁴
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The correct answer is C. Using the Power-Chain Rule: d/dx[(x² + 1)⁵] = 5(x² + 1)⁴ · d/dx[x² + 1] = 5(x² + 1)⁴ · 2x = 10x(x² + 1)⁴.

Concept Tested: Power Chain Rule

5. What is d/dx[e^(2x)]?

  1. e^(2x)
  2. 2e^(2x)
  3. 2xe^(2x)
  4. e^(2x)/2
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The correct answer is B. Using the Chain Rule: d/dx[e^(2x)] = e^(2x) · d/dx[2x] = e^(2x) · 2 = 2e^(2x).

Concept Tested: Exponential Chain Rule

6. What is d/dx[ln(x²)]?

  1. 1/x²
  2. 2/x
  3. 2x/x²
  4. 2ln(x)
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The correct answer is B. Using the Chain Rule: d/dx[ln(x²)] = (1/x²) · 2x = 2x/x² = 2/x. Alternatively, ln(x²) = 2ln(x), so d/dx = 2/x.

Concept Tested: Log Chain Rule

7. What is d/dx[cos²(x)]?

  1. 2cos(x)
  2. −2cos(x)sin(x)
  3. −sin²(x)
  4. 2cos(x)sin(x)
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The correct answer is B. Rewrite as [cos(x)]² and use Chain Rule: d/dx[cos²(x)] = 2cos(x) · (−sin(x)) = −2cos(x)sin(x). This equals −sin(2x).

Concept Tested: Nested Chain Rule

8. In Leibniz notation, if y = f(u) and u = g(x), the Chain Rule is written as:

  1. dy/dx = dy/du + du/dx
  2. dy/dx = (dy/du) · (du/dx)
  3. dy/dx = dy/du − du/dx
  4. dy/dx = (du/dx) / (dy/du)
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The correct answer is B. In Leibniz notation: dy/dx = (dy/du) · (du/dx). The du's appear to "cancel," making this notation intuitive for Chain Rule problems.

Concept Tested: Leibniz Chain Rule

9. What is d/dx[√(1 − x²)]?

  1. 1/(2√(1 − x²))
  2. −x/√(1 − x²)
  3. −2x/√(1 − x²)
  4. x/√(1 − x²)
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The correct answer is B. Rewrite as (1 − x²)^(1/2). Using Chain Rule: (1/2)(1 − x²)^(−1/2) · (−2x) = −x/√(1 − x²).

Concept Tested: General Power Rule

10. What is d/dx[sin(cos(x))]?

  1. cos(cos(x))
  2. −sin(x)cos(cos(x))
  3. cos(−sin(x))
  4. −cos(cos(x))sin(x)
Show Answer

The correct answer is D. This requires nested Chain Rule: d/dx[sin(cos(x))] = cos(cos(x)) · d/dx[cos(x)] = cos(cos(x)) · (−sin(x)) = −cos(cos(x))sin(x).

Concept Tested: Multiple Chain Rule