What's the difference between basic, intermediate, and advanced derivative problems?

Basic problems cover single-rule differentiation. Intermediate problems combine two rules together. Advanced problems involve multi-step differentiation requiring you to identify which rules apply and in what order, matching AP Calculus and university exam difficulty.

Derivative Practice Problems

Q: Is there a printable derivative practice worksheet with answers?

Yes, free downloadable PDF practice worksheets with full answer keys are available on the resources page, separate from the interactive problems on this page.

Q: How do I check if my derivative practice answers are correct?

Each problem includes a step-by-step solution reveal. For custom problems, CalcMentor's AI solver cross-verifies answers using three independent computation engines.

📅 Updated: July 12, 2026 ⏱️ 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!

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🌱 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))

⚠️ Common Mistakes While Practicing Derivatives

Even strong students lose points on the same handful of errors. Watch for these while you work through the problems above:

Forgetting the chain rule on composite functions like sin(x²) — differentiate the outer function, then multiply by the derivative of the inner function.
Dropping the minus sign when differentiating cos(x) — the derivative is −sin(x), not sin(x).
Misapplying the quotient rule order — it's (low·d(high) − high·d(low)) / low², and the order matters.
Treating e^x like a power rule problem — its derivative is itself, e^x, not x·e^(x−1).

See the full guide to common derivative mistakes →

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❓ Frequently Asked Questions

How many derivative practice problems should I solve to master calculus?

Most students need 20–40 varied problems across power rule, chain rule, product rule, quotient rule, and trig derivatives before a topic feels automatic. Work through all three difficulty levels above in order — basic, intermediate, then advanced — rather than jumping straight to hard problems.

Is there a printable derivative practice worksheet with answers?

Yes. Visit our free worksheets page for downloadable PDF practice sets with full answer keys, separate from the interactive problems on this page.

What's the difference between the basic, intermediate, and advanced levels?

Level 1 covers single-rule problems (power rule, basic trig). Level 2 combines two rules together (like chain rule with product rule). Level 3 involves multi-step problems requiring you to identify which rules apply and in what order — the same complexity level tested on AP Calculus and university exams.

How do I check if my derivative practice answers are correct?

Click "Show Solution" on any problem above for a full step-by-step breakdown. For a problem not listed here, use our CalcMentor AI solver, which cross-verifies every answer using three independent computation engines before showing you the result.

What are the most common mistakes students make in derivative practice?

Forgetting the chain rule on composite functions, dropping the negative sign on cos(x)'s derivative, and misordering the quotient rule are the three most frequent errors. See our full common mistakes guide for worked corrections.

📚 Keep Learning

🎯 Next Steps

Review Solved Examples for detailed solutions
Master All Derivative Formulas
Study Chain Rule Guide
Practice Implicit Differentiation