Unit 2 Practice Test

Differentiation - 15 Multiple Choice + 6 Free Response Questions

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Recommended time: 45 minutes

📋 Test Instructions

💡 Important: This practice test covers all topics from Unit 2: Differentiation. Take your time and show all your work for free-response questions.
  • Multiple Choice: Select the best answer for each question
  • Free Response: Show all steps and justify your reasoning
  • Time Limit: 45 minutes recommended (AP exam timing)
  • Calculator: Graphing calculator allowed for some questions

📝 Section A: Multiple Choice (15 questions)

1
Find the derivative of f(x) = x⁴ - 3x² + 2x - 1
2
Find the derivative of f(x) = (x² + 1)(x³ - 2)
3
Find the derivative of f(x) = (2x + 1)⁴
4
Find the derivative of f(x) = sin(x²)
5
Find the derivative of f(x) = e^(3x)
6
Find the derivative of f(x) = ln(x² + 1)
7
Find the derivative of f(x) = (x² + 1)/(x - 1)
8
Find the derivative of f(x) = √(x³ + 1)
9
Find the derivative of f(x) = x²e^x
10
Find the derivative of f(x) = cos(2x)
11
Find the derivative of f(x) = 1/x³
12
Find the derivative of f(x) = (x + 1)²(x - 2)
13
Find the derivative of f(x) = tan(x)
14
Find the derivative of f(x) = x²ln(x)
15
Find the derivative of f(x) = e^(x²)

✍️ Section B: Free Response (6 questions)

1
Find the derivative of f(x) = (x² + 3x)⁴ using the chain rule.

Solution:

1) Let g(x) = x² + 3x (inside function)

2) Let f(u) = u⁴ (outside function)

3) g'(x) = 2x + 3

4) f'(u) = 4u³

5) Using chain rule: f'(x) = f'(g(x)) · g'(x)

6) f'(x) = 4(x² + 3x)³ · (2x + 3)

Answer: 4(x² + 3x)³(2x + 3)

2
Find the derivative of f(x) = (x² + 1)/(x - 2) using the quotient rule.

Solution:

1) Let f(x) = x² + 1 and g(x) = x - 2

2) f'(x) = 2x and g'(x) = 1

3) Using quotient rule: d/dx[f(x)/g(x)] = [f'(x)g(x) - f(x)g'(x)]/[g(x)]²

4) = [(2x)(x - 2) - (x² + 1)(1)]/(x - 2)²

5) = [2x² - 4x - x² - 1]/(x - 2)²

6) = (x² - 4x - 1)/(x - 2)²

Answer: (x² - 4x - 1)/(x - 2)²

3
Find the derivative of f(x) = sin(x² + 1) using the chain rule.

Solution:

1) Let g(x) = x² + 1 (inside function)

2) Let f(u) = sin(u) (outside function)

3) g'(x) = 2x

4) f'(u) = cos(u)

5) Using chain rule: f'(x) = f'(g(x)) · g'(x)

6) f'(x) = cos(x² + 1) · 2x

Answer: 2x cos(x² + 1)

4
Find the derivative of f(x) = x²e^(3x) using the product rule.

Solution:

1) Let f(x) = x² and g(x) = e^(3x)

2) f'(x) = 2x

3) g'(x) = 3e^(3x) (using chain rule)

4) Using product rule: d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)

5) = (2x)(e^(3x)) + (x²)(3e^(3x))

6) = e^(3x)(2x + 3x²)

Answer: e^(3x)(2x + 3x²)

5
Find the derivative of f(x) = ln(x² + 2x + 1) using the chain rule.

Solution:

1) Let g(x) = x² + 2x + 1 (inside function)

2) Let f(u) = ln(u) (outside function)

3) g'(x) = 2x + 2

4) f'(u) = 1/u

5) Using chain rule: f'(x) = f'(g(x)) · g'(x)

6) f'(x) = (1/(x² + 2x + 1)) · (2x + 2)

7) = (2x + 2)/(x² + 2x + 1)

8) = 2(x + 1)/(x + 1)² = 2/(x + 1)

Answer: 2/(x + 1)

6
Find the derivative of f(x) = (x³ + 1)²(x² - 1) using the product rule.

Solution:

1) Let f(x) = (x³ + 1)² and g(x) = x² - 1

2) First find f'(x) using chain rule: f'(x) = 2(x³ + 1) · 3x² = 6x²(x³ + 1)

3) g'(x) = 2x

4) Using product rule: d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)

5) = [6x²(x³ + 1)](x² - 1) + (x³ + 1)²(2x)

6) = 6x²(x³ + 1)(x² - 1) + 2x(x³ + 1)²

7) = (x³ + 1)[6x²(x² - 1) + 2x(x³ + 1)]

8) = (x³ + 1)[6x⁴ - 6x² + 2x⁴ + 2x]

9) = (x³ + 1)(8x⁴ - 6x² + 2x)

Answer: (x³ + 1)(8x⁴ - 6x² + 2x)

📊 Test Results

📚 Study Recommendations

If you scored 80%+:
  • Great job! You have a solid foundation
  • Focus on the few topics you missed
  • Move on to Unit 3: Applications of Derivatives
  • Practice more challenging problems
If you scored below 80%:
  • Review the concepts you struggled with
  • Practice more basic differentiation problems
  • Focus on chain rule and product/quotient rules
  • Use the main Unit 2 page for review