Unit 3 Practice Test

Composite, Implicit, and Inverse Functions - 15 Multiple Choice + 6 Free Response Questions

← Back to Unit 3
⏱️ Test Timer
00:00

Recommended time: 45 minutes

📋 Test Instructions

💡 Important: This practice test covers all topics from Unit 3: Composite, Implicit, and Inverse Functions. 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) = (x² + 3)⁴
2
Find the derivative of f(x) = sin(x² + 1)
3
Find dy/dx for x² + y² = 25
4
Find the derivative of f(x) = e^(x²)
5
Find the derivative of f(x) = ln(x² + 1)
6
Find dy/dx for x³ + y³ = 3xy
7
Find the derivative of f(x) = arcsin(2x)
8
Find the derivative of f(x) = (x² + 1)³(2x - 1)²
9
Find the derivative of f(x) = cos(3x² + 1)
10
Find dy/dx for xy + x²y² = 6
11
Find the derivative of f(x) = arctan(x²)
12
Find the derivative of f(x) = (x + 1)²e^(x²)
13
Find dy/dx for x²y + y²x = 12
14
Find the derivative of f(x) = ln(sin(x²))
15
Find the derivative of f(x) = (x² + 1)^(x²)

✍️ Section B: Free Response (6 questions)

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

Solution:

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

2) Let f(u) = u⁵ (outside function)

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

4) f'(u) = 5u⁴

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

6) f'(x) = 5(x² + 2x)⁴ · (2x + 2)

7) = 5(x² + 2x)⁴ · 2(x + 1)

8) = 10(x + 1)(x² + 2x)⁴

Answer: 10(x + 1)(x² + 2x)⁴

2
Find dy/dx for x²y + y³ = 8 using implicit differentiation.

Solution:

1) Differentiate both sides with respect to x:

2) d/dx[x²y] + d/dx[y³] = d/dx[8]

3) Use product rule for x²y: 2xy + x² · dy/dx

4) Chain rule for y³: 3y² · dy/dx

5) So: 2xy + x² · dy/dx + 3y² · dy/dx = 0

6) Factor out dy/dx: dy/dx(x² + 3y²) = -2xy

7) Solve for dy/dx: dy/dx = -2xy/(x² + 3y²)

Answer: dy/dx = -2xy/(x² + 3y²)

3
Find the derivative of f(x) = arcsin(3x) using the chain rule.

Solution:

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

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

3) g'(x) = 3

4) f'(u) = 1/√(1 - u²)

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

6) f'(x) = 1/√(1 - (3x)²) · 3

7) = 3/√(1 - 9x²)

Answer: 3/√(1 - 9x²)

4
Find the derivative of f(x) = e^(x² + 1) · ln(x² + 1) using the product rule.

Solution:

1) Let u = e^(x² + 1) and v = ln(x² + 1)

2) Find u': u' = e^(x² + 1) · 2x = 2xe^(x² + 1)

3) Find v': v' = 2x/(x² + 1)

4) Using product rule: f'(x) = u'v + uv'

5) = 2xe^(x² + 1) · ln(x² + 1) + e^(x² + 1) · 2x/(x² + 1)

6) = e^(x² + 1)[2x ln(x² + 1) + 2x/(x² + 1)]

7) = 2xe^(x² + 1)[ln(x² + 1) + 1/(x² + 1)]

Answer: 2xe^(x² + 1)[ln(x² + 1) + 1/(x² + 1)]

5
Find dy/dx for x³ + y³ = 3xy using implicit differentiation.

Solution:

1) Differentiate both sides with respect to x:

2) 3x² + 3y² · dy/dx = 3y + 3x · dy/dx

3) Collect dy/dx terms: 3y² · dy/dx - 3x · dy/dx = 3y - 3x²

4) Factor: dy/dx(3y² - 3x) = 3y - 3x²

5) Factor out 3: dy/dx(3)(y² - x) = 3(y - x²)

6) Cancel 3: dy/dx(y² - x) = y - x²

7) Solve: dy/dx = (y - x²)/(y² - x)

Answer: dy/dx = (y - x²)/(y² - x)

6
Find the derivative of f(x) = (x² + 1)^(x) using logarithmic differentiation.

Solution:

1) Let y = (x² + 1)^x

2) Take natural log: ln(y) = ln((x² + 1)^x) = x ln(x² + 1)

3) Differentiate implicitly: (1/y) · dy/dx = ln(x² + 1) + x · 2x/(x² + 1)

4) Simplify: (1/y) · dy/dx = ln(x² + 1) + 2x²/(x² + 1)

5) Solve for dy/dx: dy/dx = y[ln(x² + 1) + 2x²/(x² + 1)]

6) Substitute back: dy/dx = (x² + 1)^x[ln(x² + 1) + 2x²/(x² + 1)]

Answer: (x² + 1)^x[ln(x² + 1) + 2x²/(x² + 1)]

📊 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 4: Applications of Derivatives
  • Practice more challenging problems
If you scored below 80%:
  • Review the concepts you struggled with
  • Practice more implicit differentiation problems
  • Focus on chain rule and inverse functions
  • Use the main Unit 3 page for review