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Quiz: Product, Quotient, and Transcendental Derivatives

Test your understanding of derivative rules for products, quotients, and transcendental functions.

1. The Product Rule states that d/dx[f(x)g(x)] equals:

  1. f'(x) · g'(x)
  2. f'(x)g(x) + f(x)g'(x)
  3. f(x)g'(x) − f'(x)g(x)
  4. [f(x)g(x)]'
Show Answer

The correct answer is B. The Product Rule: d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x). "Derivative of first times second, plus first times derivative of second."

Concept Tested: Product Rule

2. What is d/dx[x² · sin(x)]?

  1. 2x · cos(x)
  2. 2x · sin(x) + x² · cos(x)
  3. x² · cos(x)
  4. 2x · cos(x) − x² · sin(x)
Show Answer

The correct answer is B. Using the Product Rule: d/dx[x²·sin(x)] = (2x)·sin(x) + x²·cos(x). Let f = x², g = sin(x), then f' = 2x, g' = cos(x).

Concept Tested: Product Rule Formula

3. The Quotient Rule states that d/dx[f(x)/g(x)] equals:

  1. f'(x)/g'(x)
  2. [f'(x)g(x) + f(x)g'(x)]/[g(x)]²
  3. [f'(x)g(x) − f(x)g'(x)]/[g(x)]²
  4. [f(x)g'(x) − f'(x)g(x)]/[g(x)]²
Show Answer

The correct answer is C. The Quotient Rule: d/dx[f/g] = [f'g − fg']/g². "Low d-high minus high d-low, over low squared."

Concept Tested: Quotient Rule

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

  1. −cos(x)
  2. cos(x)
  3. −sin(x)
  4. tan(x)
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The correct answer is B. The derivative of sine is cosine: d/dx[sin(x)] = cos(x). This is one of the fundamental trig derivatives that must be memorized.

Concept Tested: Derivative of Sine

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

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

The correct answer is B. The derivative of cosine is negative sine: d/dx[cos(x)] = −sin(x). Notice the pattern: sine → cosine → −sine → −cosine → sine.

Concept Tested: Derivative of Cosine

6. What is d/dx[tan(x)]?

  1. sec(x)
  2. sec²(x)
  3. cot(x)
  4. −csc²(x)
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The correct answer is B. The derivative of tangent is secant squared: d/dx[tan(x)] = sec²(x). This can be derived using the Quotient Rule on sin(x)/cos(x).

Concept Tested: Derivative of Tangent

7. What is d/dx[eˣ]?

  1. xeˣ⁻¹
  2. eˣ
  3. eˣ⁻¹
  4. ln(x)·eˣ
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The correct answer is B. The natural exponential function is its own derivative: d/dx[eˣ] = eˣ. This remarkable property makes e the natural base for calculus.

Concept Tested: Derivative of e to x

8. What is d/dx[ln(x)]?

  1. 1/x
  2. ln(x)/x
  3. x
  4. eˣ
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The correct answer is A. The derivative of the natural logarithm is the reciprocal function: d/dx[ln(x)] = 1/x (for x > 0). This is essential for integration.

Concept Tested: Derivative of ln x

9. What is d/dx[x/sin(x)] using the Quotient Rule?

  1. cos(x)/sin²(x)
  2. [sin(x) − x·cos(x)]/sin²(x)
  3. 1/cos(x)
  4. [sin(x) + x·cos(x)]/sin²(x)
Show Answer

The correct answer is B. Using the Quotient Rule with f = x, g = sin(x): d/dx[x/sin(x)] = [(1)·sin(x) − x·cos(x)]/sin²(x) = [sin(x) − x·cos(x)]/sin²(x).

Concept Tested: Quotient Rule Formula

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

  1. sec(x)tan(x)
  2. −sec(x)tan(x)
  3. csc(x)cot(x)
  4. sec²(x)
Show Answer

The correct answer is A. The derivative of secant is secant times tangent: d/dx[sec(x)] = sec(x)tan(x). This comes from the Quotient Rule applied to 1/cos(x).

Concept Tested: Derivative of Secant