Derivatives Quiz

Grade 12 · Math · Quiz

10 questions on differentiation rules and techniques for 12th grade math.

Questions

1. What is d/dx[x⁵]?

   1. x⁴
   2. 5x⁴
   3. 5x⁵
   4. 4x⁴
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   Answer: 5x⁴

   Power rule: d/dx[xⁿ] = nxⁿ⁻¹. So d/dx[x⁵] = 5x⁴.

2. Using the quotient rule, find d/dx[x²/sin x].

   1. (2x sin x - x² cos x)/sin²x
   2. (2x cos x + x² sin x)/sin²x
   3. (2x sin x + x² cos x)/cos²x
   4. 2x/cos x
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   Answer: (2x sin x - x² cos x)/sin²x

   Quotient rule: (f′g - fg′)/g² = (2x·sin x - x²·cos x)/sin²x.

3. Find f′(x) if f(x) = (3x + 1)⁵.

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   Chain rule: 5(3x+1)⁴ · 3 = 15(3x+1)⁴.

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

   1. sin x
   2. −sin x
   3. cos x
   4. −cos x
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   Answer: −sin x

   Standard derivative: d/dx[cos x] = −sin x.

5. The derivative of eˣ is xeˣ⁻¹.

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   The derivative of eˣ is eˣ, not xeˣ⁻¹. The power rule does not apply to exponential functions.

6. What is the slope of the tangent to f(x) = x³ - 3x at x = 2?

   1. 3
   2. 6
   3. 9
   4. 2
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   Answer: 9

   f′(x) = 3x² - 3. At x=2: f′(2) = 3(4) - 3 = 12 - 3 = 9.

7. Find dy/dx if y = ln(x² + 1).

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   Chain rule: d/dx[ln(u)] = (1/u)·u′. u = x²+1, u′ = 2x. Result: 2x/(x²+1).

8. Which rule applies to d/dx[sin(x)·cos(x)]?

   1. Chain rule
   2. Power rule
   3. Product rule
   4. Quotient rule
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   Answer: Product rule

   sin(x)·cos(x) is a product of two functions, so we use the product rule.

9. d/dx[7] = 7.

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   The derivative of any constant is 0. d/dx[7] = 0.

10. Find d/dx[tan x].

    Show answer

    tan x = sin x/cos x. Quotient rule gives (cos²x + sin²x)/cos²x = 1/cos²x = sec²x.

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