Derivative Practice Problems

📅 Updated: November 28, 2024 📊 15+ Detailed Examples 🧮 Calculator Integration

Master derivative practice problems with comprehensive examples featuring detailed step-by-step solutions and integrated calculator tools for instant verification. Each problem includes complete working, explanations, and links to specialized calculators. Perfect for homework help, exam preparation, and building problem-solving skills.

💡 How to Use This Guide

Work through each problem on paper first, then check your solution against ours. Use the integrated calculators to verify your answers and see alternative solution methods. Start with easier problems and progress to advanced!

📑 Problem Categories

  1. Power Rule Problems
  2. Product Rule Problems
  3. Chain Rule Problems
  4. Quotient Rule Problems
  5. Mixed Advanced Problems

🌱 Power Rule Practice Problems

Master the fundamental power rule with these detailed examples. Review our formula reference if needed!

📐 Power Rule Formula

d/dx[x^n] = n·x^(n-1)

This works for all real numbers n, including negative and fractional exponents.

📝 Example 1: Polynomial Function EASY
Find: f'(x) if f(x) = 3x⁴ - 2x³ + 5x² - 7x + 9
1 Apply power rule to each term separately:

d/dx[3x⁴] = 3 · 4x³ = 12x³

d/dx[-2x³] = -2 · 3x² = -6x²

d/dx[5x²] = 5 · 2x = 10x

d/dx[-7x] = -7 · 1 = -7

d/dx[9] = 0 (constant rule)

2 Combine all terms:

f'(x) = 12x³ - 6x² + 10x - 7

Answer: f'(x) = 12x³ - 6x² + 10x - 7
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📝 Example 2: Negative & Fractional Exponents MEDIUM
Find: d/dx[2/x³ + √x - 4/√x]
1 Rewrite using exponents:

2/x³ = 2x^(-3)

√x = x^(1/2)

4/√x = 4x^(-1/2)

So: f(x) = 2x^(-3) + x^(1/2) - 4x^(-1/2)

2 Apply power rule to each term:

d/dx[2x^(-3)] = 2(-3)x^(-4) = -6x^(-4) = -6/x⁴

d/dx[x^(1/2)] = (1/2)x^(-1/2) = 1/(2√x)

d/dx[-4x^(-1/2)] = -4(-1/2)x^(-3/2) = 2x^(-3/2) = 2/(x√x)

Answer: f'(x) = -6/x⁴ + 1/(2√x) + 2/(x√x)
💡 Pro Tip

Always rewrite radicals and fractions as powers first! This makes applying the power rule much easier.

⚡ Product Rule Practice Problems

Practice the product rule with these examples. See our complete Product Rule Tutorial for more!

📐 Product Rule Formula

(uv)' = u'v + uv'

Memory aid: "First times derivative of second, plus second times derivative of first"

📝 Example 3: Polynomial Product MEDIUM
Find: d/dx[(x² + 3)(2x³ - 5)]
1 Identify u and v:

Let u = x² + 3

Let v = 2x³ - 5

2 Find u' and v':

u' = 2x

v' = 6x²

3 Apply product rule (uv)' = u'v + uv':

= (2x)(2x³ - 5) + (x² + 3)(6x²)

= 4x⁴ - 10x + 6x⁴ + 18x²

= 10x⁴ + 18x² - 10x

Answer: f'(x) = 10x⁴ + 18x² - 10x
🧮 Product Rule Calculator

Verify with our specialized product rule calculator:

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📝 Example 4: Exponential Product HARD
Find: d/dx[x³·e^x]
1 Set up product rule:

u = x³, u' = 3x²

v = e^x, v' = e^x (special property!)

2 Apply formula:

= 3x²·e^x + x³·e^x

= e^x(3x² + x³)

= e^x·x²(3 + x)

Answer: f'(x) = e^x·x²(x + 3)
⚠️ Common Mistake

Wrong: (uv)' = u'v'

Right: (uv)' = u'v + uv'

You must use BOTH terms, not just multiply the derivatives!

🔗 Chain Rule Practice Problems

Master composite functions with the chain rule. Check our Chain Rule Complete Guide!

📐 Chain Rule Formula

d/dx[f(g(x))] = f'(g(x))·g'(x)

Or: (outer derivative) × (inner derivative)

📝 Example 5: Power Chain MEDIUM
Find: d/dx[(5x² - 3x + 1)⁶]
1 Identify outer and inner functions:

Outer function: f(u) = u⁶

Inner function: g(x) = 5x² - 3x + 1

2 Differentiate outer (keep inner unchanged):

f'(u) = 6u⁵

So: 6(5x² - 3x + 1)⁵

3 Multiply by derivative of inner:

g'(x) = 10x - 3

Result: 6(5x² - 3x + 1)⁵·(10x - 3)

Answer: f'(x) = 6(10x - 3)(5x² - 3x + 1)⁵
🧮 Chain Rule Calculator

Verify complex chain rule problems:

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📝 Example 6: Triple Chain Rule HARD
Find: d/dx[sin(cos(x²))]
1 Identify three layers:

Outermost: sin( )

Middle: cos( )

Innermost: x²

2 Apply chain rule three times:

Derivative of sin(u) = cos(u), where u = cos(x²)

So far: cos(cos(x²))·...

3 Continue with inner layers:

Derivative of cos(v) = -sin(v), where v = x²

So far: cos(cos(x²))·(-sin(x²))·...

4 Complete with innermost:

Derivative of x² = 2x

Full result: cos(cos(x²))·(-sin(x²))·2x

Answer: f'(x) = -2x·sin(x²)·cos(cos(x²))
💡 Strategy for Nested Chains

Work from outside to inside, one layer at a time. Write each derivative as you go, then multiply them all together at the end!

➗ Quotient Rule Practice Problems

Practice differentiating rational functions with the quotient rule.

📐 Quotient Rule Formula

(u/v)' = (u'v - uv')/v²

Memory aid: "Low d-high minus high d-low, over the square of what's below"

📝 Example 7: Basic Quotient MEDIUM
Find: d/dx[(3x² + 2)/(x - 1)]
1 Identify u and v:

u = 3x² + 2, u' = 6x

v = x - 1, v' = 1

2 Apply quotient rule:

= [(6x)(x - 1) - (3x² + 2)(1)] / (x - 1)²

= [6x² - 6x - 3x² - 2] / (x - 1)²

= (3x² - 6x - 2) / (x - 1)²

Answer: f'(x) = (3x² - 6x - 2)/(x - 1)²
🧮 Quotient Rule Calculator

Check quotient rule problems with step-by-step solutions:

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📝 Example 8: Trigonometric Quotient HARD
Find: d/dx[sin(x)/x²]
1 Set up quotient rule:

u = sin(x), u' = cos(x)

v = x², v' = 2x

2 Apply formula (u'v - uv')/v²:

= [cos(x)·x² - sin(x)·2x] / (x²)²

= [x²cos(x) - 2x·sin(x)] / x⁴

3 Simplify by factoring:

= [x(x·cos(x) - 2sin(x))] / x⁴

= [x·cos(x) - 2sin(x)] / x³

Answer: f'(x) = [x·cos(x) - 2sin(x)]/x³

🚀 Mixed Advanced Problems

Challenge yourself with problems requiring multiple rules! Also try Implicit Differentiation.

📝 Example 9: Product + Chain Rule HARD
Find: d/dx[e^(x²)·sin(3x)]
1 Recognize this needs product rule:

u = e^(x²), v = sin(3x)

Both u and v require chain rule for their derivatives

2 Find u' using chain rule:

u' = e^(x²)·2x = 2x·e^(x²)

3 Find v' using chain rule:

v' = cos(3x)·3 = 3cos(3x)

4 Apply product rule (uv)' = u'v + uv':

= (2x·e^(x²))·sin(3x) + e^(x²)·3cos(3x)

= 2x·e^(x²)·sin(3x) + 3e^(x²)·cos(3x)

= e^(x²)[2x·sin(3x) + 3cos(3x)]

Answer: f'(x) = e^(x²)[2x·sin(3x) + 3cos(3x)]
💡 Strategy

When combining rules, work methodically: Identify which main rule applies first (product, quotient, or chain), then apply additional rules as needed for each piece.

📝 Example 10: Quotient + Chain Rule HARD
Find: d/dx[ln(x²)/√x]
1 Identify quotient and rewrite:

u = ln(x²), v = √x = x^(1/2)

2 Find u' using chain rule:

u' = (1/x²)·2x = 2/x

3 Find v' using power rule:

v' = (1/2)x^(-1/2) = 1/(2√x)

4 Apply quotient rule:

= [(2/x)·√x - ln(x²)·1/(2√x)] / x

= [2√x/x - ln(x²)/(2√x)] / x

= [4x - ln(x²)] / (2x√x)

Answer: f'(x) = [4x - ln(x²)]/(2x√x)

🧮 Master Derivatives with Our Calculators

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