Derivative Practice Problems

📅 Updated: November 28, 2024 ⏱️ 30 min practice 🎯 30 Problems

Test your skills with comprehensive derivative practice problems from basic to advanced. Challenge yourself with over 30 calculus exercises covering power rule, chain rule, product rule, quotient rule, trigonometric derivatives, and implicit differentiation. Try solving each problem before revealing the solution!

💡 How to Practice Effectively

Work through problems on paper first, then click "Show Solution" to check your work. This active practice approach maximizes learning and retention!

📑 Problem Sets

  1. Level 1: Basic (10 problems)
  2. Level 2: Intermediate (10 problems)
  3. Level 3: Advanced (10 problems)

🌱 Level 1: Basic Problems

Practice fundamental derivative rules including power rule and basic combinations.

Problem 1 EASY
Find: d/dx[x⁵]
✅ Solution:

Apply power rule: d/dx[x^n] = n·x^(n-1)

f'(x) = 5x⁴
Problem 2 EASY
Find: d/dx[7x³ - 4x² + 9x - 2]
✅ Solution:

Apply power rule to each term

f'(x) = 21x² - 8x + 9
Problem 3 EASY
Find: d/dx[√x]
✅ Solution:

Rewrite as x^(1/2), then apply power rule

f'(x) = 1/(2√x) or (1/2)x^(-1/2)
Problem 4 EASY
Find: d/dx[1/x³]
✅ Solution:

Rewrite as x^(-3), apply power rule

f'(x) = -3/x⁴ or -3x^(-4)
Problem 5 EASY
Find: d/dx[sin(x)]
✅ Solution:

Basic trig derivative

f'(x) = cos(x)
Problem 6 EASY
Find: d/dx[e^x + ln(x)]
✅ Solution:

d/dx[e^x] = e^x, d/dx[ln(x)] = 1/x

f'(x) = e^x + 1/x
Problem 7 EASY
Find: d/dx[5x⁴ - 3x² + 7]
✅ Solution:

Apply power rule, constant becomes 0

f'(x) = 20x³ - 6x
Problem 8 EASY
Find: d/dx[cos(x)]
✅ Solution:

Trig derivative with negative sign

f'(x) = -sin(x)
Problem 9 EASY
Find: d/dx[x^(2/3)]
✅ Solution:

Power rule with fractional exponent

f'(x) = (2/3)x^(-1/3)
Problem 10 EASY
Find: d/dx[tan(x)]
✅ Solution:

Trig derivative

f'(x) = sec²(x)

⚡ Level 2: Intermediate Problems

Challenge yourself with product rule, chain rule, and quotient rule practice. Review Chain Rule and Product Rule if needed!

Problem 11 MEDIUM
Find: d/dx[(x² + 1)(x³ - 2)]
✅ Solution:

Product rule: (uv)' = u'v + uv'

f'(x) = 2x(x³ - 2) + (x² + 1)(3x²) = 5x⁴ + 3x² - 4x
Problem 12 MEDIUM
Find: d/dx[(2x + 1)⁴]
✅ Solution:

Chain rule: outer derivative × inner derivative

f'(x) = 4(2x + 1)³ · 2 = 8(2x + 1)³
Problem 13 MEDIUM
Find: d/dx[x²·sin(x)]
✅ Solution:

Product rule with trig

f'(x) = 2x·sin(x) + x²·cos(x)
Problem 14 MEDIUM
Find: d/dx[sin(3x²)]
✅ Solution:

Chain rule: cos(3x²) × 6x

f'(x) = 6x·cos(3x²)
Problem 15 MEDIUM
Find: d/dx[(x³ + 1)/(x - 2)]
✅ Solution:

Quotient rule: (u'v - uv')/v²

f'(x) = (3x²(x - 2) - (x³ + 1))/(x - 2)² = (2x³ - 6x² - 1)/(x - 2)²
Problem 16 MEDIUM
Find: d/dx[e^(2x)]
✅ Solution:

Chain rule with exponential

f'(x) = 2e^(2x)
Problem 17 MEDIUM
Find: d/dx[ln(x² + 1)]
✅ Solution:

Chain rule: (1/(x² + 1)) × 2x

f'(x) = 2x/(x² + 1)
Problem 18 MEDIUM
Find: d/dx[x·e^x]
✅ Solution:

Product rule: u' = 1, v' = e^x

f'(x) = e^x + x·e^x = e^x(1 + x)
Problem 19 MEDIUM
Find: d/dx[√(x² + 4)]
✅ Solution:

Chain rule: (1/2)(x² + 4)^(-1/2) × 2x

f'(x) = x/√(x² + 4)
Problem 20 MEDIUM
Find: d/dx[cos(x²)]
✅ Solution:

Chain rule: -sin(x²) × 2x

f'(x) = -2x·sin(x²)

🚀 Level 3: Advanced Problems

Master advanced derivative techniques including nested chains, multiple rules, and implicit differentiation. Check Implicit Differentiation Guide!

Problem 21 HARD
Find: d/dx[sin(cos(x²))]
✅ Solution:

Triple chain rule

f'(x) = -2x·sin(x²)·cos(cos(x²))
Problem 22 HARD
Find dy/dx: x² + y² = 25
✅ Solution:

Implicit differentiation

dy/dx = -x/y
Problem 23 HARD
Find: d/dx[e^(x²)·sin(x)]
✅ Solution:

Product rule + chain rule

f'(x) = 2x·e^(x²)·sin(x) + e^(x²)·cos(x)
Problem 24 HARD
Find: d/dx[(x² + 1)^5·(x - 3)³]
✅ Solution:

Product rule with chain rule on both terms

f'(x) = 10x(x² + 1)⁴(x - 3)³ + 3(x² + 1)⁵(x - 3)²
Problem 25 HARD
Find dy/dx: xy + y² = 10
✅ Solution:

Implicit differentiation with product rule

dy/dx = -y/(x + 2y)
Problem 26 HARD
Find: d/dx[ln(sin(x²))]
✅ Solution:

Chain rule three times

f'(x) = 2x·cot(x²)
Problem 27 HARD
Find: d/dx[sin(x)/x²]
✅ Solution:

Quotient rule

f'(x) = (x·cos(x) - 2sin(x))/x³
Problem 28 HARD
Find: d/dx[e^(sin(x))]
✅ Solution:

Chain rule with exponential

f'(x) = cos(x)·e^(sin(x))
Problem 29 HARD
Find dy/dx: x³ + y³ = 3xy
✅ Solution:

Implicit differentiation (Folium of Descartes)

dy/dx = (y - x²)/(y² - x)
Problem 30 HARD
Find: d/dx[tan(e^(x²))]
✅ Solution:

Triple chain rule

f'(x) = 2x·e^(x²)·sec²(e^(x²))

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