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Quiz: Basic Derivative Rules

Test your understanding of basic derivative rules with these review questions.

1. What is the derivative of a constant function f(x) = 7?

  1. 7
  2. 1
  3. 0
  4. x
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The correct answer is C. The Constant Rule states that the derivative of any constant is zero. Since constants don't change, their rate of change is zero: d/dx[c] = 0.

Concept Tested: Constant Rule

2. Using the Power Rule, what is d/dx[x⁵]?

  1. 5x⁴
  2. 5x⁵
  3. x⁴
  4. 4x⁵
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The correct answer is A. The Power Rule states d/dx[xⁿ] = nxⁿ⁻¹. For x⁵: d/dx[x⁵] = 5x⁴. Bring down the exponent and reduce it by one.

Concept Tested: Power Rule Derivative

3. What is d/dx[3x⁴ − 2x² + 5x − 1]?

  1. 12x³ − 4x + 5
  2. 12x³ − 4x + 5 − 1
  3. 3x³ − 2x + 5
  4. 12x⁴ − 4x² + 5x
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The correct answer is A. Apply rules term by term: d/dx[3x⁴] = 12x³, d/dx[−2x²] = −4x, d/dx[5x] = 5, d/dx[−1] = 0. Sum: 12x³ − 4x + 5.

Concept Tested: Polynomial Derivative

4. What is the derivative of f(x) = x⁻³?

  1. −3x⁻⁴
  2. −3x⁻²
  3. 3x⁻⁴
  4. −4x⁻³
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The correct answer is A. The Power Rule works for negative exponents: d/dx[x⁻³] = −3x⁻³⁻¹ = −3x⁻⁴. This can also be written as −3/x⁴.

Concept Tested: Negative Exponent Derivative

5. What is d/dx[√x]?

  1. 1/(2√x)
  2. 2√x
  3. 1/√x
  4. √x/2
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The correct answer is A. Rewrite √x = x^(1/2), then apply the Power Rule: d/dx[x^(1/2)] = (1/2)x^(−1/2) = 1/(2√x).

Concept Tested: Derivative of Root

6. The Sum Rule for derivatives states that d/dx[f(x) + g(x)] equals:

  1. f'(x) · g'(x)
  2. f'(x) + g'(x)
  3. f(x)g'(x) + g(x)f'(x)
  4. [f'(x) + g'(x)]/(f(x) + g(x))
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The correct answer is B. The Sum Rule states that the derivative of a sum is the sum of the derivatives: d/dx[f(x) + g(x)] = f'(x) + g'(x). Derivatives are linear.

Concept Tested: Sum Rule Derivative

7. What is d/dx[5f(x)] if f'(x) = 2x?

  1. 5f(x)
  2. 10x
  3. 2x
  4. 5 + 2x
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The correct answer is B. The Constant Multiple Rule states d/dx[cf(x)] = c·f'(x). So d/dx[5f(x)] = 5·f'(x) = 5·(2x) = 10x.

Concept Tested: Constant Multiple Derivative

8. What is the derivative of f(x) = 4x³ − 7x + 2?

  1. 12x² − 7
  2. 12x² − 7x
  3. 4x² − 7
  4. 12x³ − 7
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The correct answer is A. Using the Power Rule and Sum Rule: d/dx[4x³] = 12x², d/dx[−7x] = −7, d/dx[2] = 0. Combined: 12x² − 7.

Concept Tested: Linear Combination

9. What is d/dx[x^(3/2)]?

  1. (3/2)x^(1/2)
  2. (3/2)x^(5/2)
  3. (1/2)x^(3/2)
  4. (2/3)x^(1/2)
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The correct answer is A. Apply the Power Rule with fractional exponent: d/dx[x^(3/2)] = (3/2)x^(3/2−1) = (3/2)x^(1/2) = (3/2)√x.

Concept Tested: Fractional Exponent

10. Find f'(2) if f(x) = x³ − 4x + 1.

  1. 1
  2. 5
  3. 8
  4. 12
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The correct answer is C. First find f'(x) = 3x² − 4. Then evaluate at x = 2: f'(2) = 3(4) − 4 = 12 − 4 = 8.

Concept Tested: Derivative of x to n