Solved Derivative Examples

📅 Updated: November 28, 2024 ⏱️ 25 min read 📝 20+ Problems

Master calculus with comprehensive solved derivative problems featuring detailed step-by-step solutions. From basic power rule examples to advanced chain rule, product rule, and implicit differentiation problems, each example includes complete explanations to help you understand the process and build problem-solving skills.

💡 How to Use These Examples

Try solving each problem yourself first, then check the step-by-step solution. This active practice method dramatically improves retention compared to passive reading!

🌱 Basic Examples (Power Rule)

Start with fundamental power rule examples to build your foundation.

📝 Example 1: Simple Polynomial EASY

Find: d/dx[x⁴ - 3x² + 5x - 7]

1 Apply power rule to each term:

d/dx[x⁴] = 4x³

d/dx[-3x²] = -3(2x) = -6x

d/dx[5x] = 5

d/dx[-7] = 0

2 Combine all terms:

f'(x) = 4x³ - 6x + 5

Answer: f'(x) = 4x³ - 6x + 5

📝 Example 2: Fractional Exponents EASY

Find: d/dx[√x + 1/x²]

1 Rewrite using exponents:

√x = x^(1/2)

1/x² = x^(-2)

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

2 Apply power rule:

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

d/dx[x^(-2)] = -2x^(-3) = -2/x³

Answer: f'(x) = 1/(2√x) - 2/x³

📝 Example 3: Polynomial Factored Form MEDIUM

Find: d/dx[(x - 1)(x + 2)(x - 3)]

1 Expand first:

(x - 1)(x + 2) = x² + x - 2

(x² + x - 2)(x - 3) = x³ - 2x² - 5x + 6

2 Apply power rule:

f'(x) = 3x² - 4x - 5

Answer: f'(x) = 3x² - 4x - 5

⚡ Product Rule Examples

Master the product rule with these worked examples. See the complete Product Rule Tutorial for more!

📝 Example 4: Polynomial × Trig MEDIUM

Find: d/dx[x³ · sin(x)]

1 Identify u and v:

u = x³, v = sin(x)

u' = 3x², v' = cos(x)

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

= 3x² · sin(x) + x³ · cos(x)

Answer: f'(x) = 3x²sin(x) + x³cos(x)

📝 Example 5: Exponential × Polynomial MEDIUM

Find: d/dx[e^x · (x² + 1)]

1 Set up product rule:

u = e^x, u' = e^x

v = x² + 1, v' = 2x

2 Apply formula:

= e^x · (x² + 1) + e^x · 2x

= e^x(x² + 1 + 2x)

= e^x(x² + 2x + 1)

Answer: f'(x) = e^x(x + 1)²

🔗 Chain Rule Examples

Practice chain rule problems with composite functions. Check our Chain Rule Complete Guide for detailed explanations!

📝 Example 6: Power of Polynomial MEDIUM

Find: d/dx[(3x² - 5)⁴]

1 Identify outer and inner:

Outer: ( )⁴

Inner: 3x² - 5

2 Differentiate outer (keep inner):

4(3x² - 5)³

3 Multiply by derivative of inner:

= 4(3x² - 5)³ · 6x

= 24x(3x² - 5)³

Answer: f'(x) = 24x(3x² - 5)³

📝 Example 7: Nested Trig Function HARD

Find: d/dx[sin(cos(x²))]

1 Identify three layers:

Outermost: sin( )

Middle: cos( )

Innermost: x²

2 Apply chain rule three times:

Outer derivative: cos(cos(x²))

Middle derivative: -sin(x²)

Inner derivative: 2x

3 Multiply all together:

= cos(cos(x²)) · (-sin(x²)) · 2x

Answer: f'(x) = -2x·sin(x²)·cos(cos(x²))

📝 Example 8: Exponential Chain MEDIUM

Find: d/dx[e^(x² + 3x)]

1 Identify:

Outer: e^( )

Inner: x² + 3x

2 Apply chain rule:

= e^(x² + 3x) · (2x + 3)

Answer: f'(x) = (2x + 3)e^(x² + 3x)

➗ Quotient Rule Examples

Master quotient rule differentiation with these examples.

📝 Example 9: Polynomial Quotient MEDIUM

Find: d/dx[(x² + 1)/(x - 2)]

1 Set up quotient rule:

u = x² + 1, u' = 2x

v = x - 2, v' = 1

2 Apply: (u/v)' = (u'v - uv')/v²

= [2x(x - 2) - (x² + 1)(1)]/(x - 2)²

= [2x² - 4x - x² - 1]/(x - 2)²

= (x² - 4x - 1)/(x - 2)²

Answer: f'(x) = (x² - 4x - 1)/(x - 2)²

📝 Example 10: Trig Quotient HARD

Find: d/dx[sin(x)/cos(x)] (Verify this equals sec²(x))

1 Apply quotient rule:

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

v = cos(x), v' = -sin(x)

2 Calculate:

= [cos(x)·cos(x) - sin(x)·(-sin(x))]/cos²(x)

= [cos²(x) + sin²(x)]/cos²(x)

= 1/cos²(x) = sec²(x) ✓

Answer: f'(x) = sec²(x)

📐 Trigonometric Examples

Practice trigonometric derivatives with various functions.

📝 Example 11: Trig Chain Rule MEDIUM

Find: d/dx[tan(3x²)]

1 Apply chain rule:

Outer: tan( ) → sec²( )

Inner: 3x² → 6x

2 Combine:

= sec²(3x²) · 6x

Answer: f'(x) = 6x·sec²(3x²)

📝 Example 12: Product of Trig HARD

Find: d/dx[x·sin(x)·cos(x)]

1 Use product rule twice:

Let u = x·sin(x), v = cos(x)

First find u': u' = sin(x) + x·cos(x)

2 Apply product rule:

= u'v + uv'

= [sin(x) + x·cos(x)]·cos(x) + x·sin(x)·(-sin(x))

= sin(x)cos(x) + x·cos²(x) - x·sin²(x)

Answer: f'(x) = sin(x)cos(x) + x·cos(2x)

🔄 Implicit Differentiation Examples

Master implicit differentiation for equations not solved for y. See our Implicit Differentiation Guide!

📝 Example 13: Circle Equation MEDIUM

Find dy/dx: x² + y² = 25

1 Differentiate both sides:

d/dx[x²] + d/dx[y²] = d/dx[25]

2x + 2y·(dy/dx) = 0

2 Solve for dy/dx:

2y·(dy/dx) = -2x

dy/dx = -2x/2y = -x/y

Answer: dy/dx = -x/y

📝 Example 14: Folium of Descartes HARD

Find dy/dx: x³ + y³ = 6xy

1 Differentiate (use product rule on right):

3x² + 3y²·(dy/dx) = 6y + 6x·(dy/dx)

2 Collect dy/dx terms:

3y²·(dy/dx) - 6x·(dy/dx) = 6y - 3x²

(3y² - 6x)·(dy/dx) = 6y - 3x²

3 Solve:

dy/dx = (6y - 3x²)/(3y² - 6x)

dy/dx = (2y - x²)/(y² - 2x)

Answer: dy/dx = (2y - x²)/(y² - 2x)

🎯 Practice More Problems

Test your skills with our interactive derivative calculator featuring step-by-step solutions!

Try Derivative Calculator →

📚 Study Tips

🎯 How to Master Derivatives

Practice actively: Try problems before checking solutions
Focus on patterns: Notice when to use each rule
Check your work: Use our calculator to verify answers
Review formulas: Keep our formula reference handy
Build gradually: Master basic problems before advanced ones

📖 Related Resources

Continue learning with these detailed guides: