Product Rule for Derivatives

Learn how to find derivatives of products of functions with step-by-step examples.

The Product Rule Formula

If f(x) = u(x) · v(x), then f'(x) = u'(x) · v(x) + u(x) · v'(x)

Or simply: (uv)' = u'v + uv'

📝 Memory Aid:

"First times derivative of second, plus second times derivative of first"

• u'v: derivative of first × second
• uv': first × derivative of second

🎯 When to use:

• Two functions multiplied together
• x² · sin(x)
• e^x · ln(x)
• (x+1) · (x²-3)

Step-by-Step Examples

Example 1: Polynomial Product

Find: d/dx[x² · (x + 3)]

Step 1: Identify u = x², v = (x + 3)

Step 2: Find u' = 2x, v' = 1

Step 3: Apply product rule: u'v + uv'

Step 4: 2x(x + 3) + x²(1)

Step 5: 2x² + 6x + x² = 3x² + 6x

Answer: 3x² + 6x

Example 2: Trigonometric Product

Find: d/dx[x · sin(x)]

Step 1: Identify u = x, v = sin(x)

Step 2: Find u' = 1, v' = cos(x)

Step 3: Apply product rule: u'v + uv'

Step 4: 1 · sin(x) + x · cos(x)

Answer: sin(x) + x cos(x)

Example 3: Exponential Product

Find: d/dx[x² · e^x]

Step 1: Identify u = x², v = e^x

Step 2: Find u' = 2x, v' = e^x

Step 3: Apply product rule: u'v + uv'

Step 4: 2x · e^x + x² · e^x

Step 5: Factor: e^x(2x + x²)

Answer: e^x(x² + 2x)

⚠️ Common Mistakes to Avoid

❌ Wrong

d/dx[x² · sin(x)] = 2x · cos(x)

This just multiplies the derivatives - missing the product rule!

✅ Correct

d/dx[x² · sin(x)] = 2x · sin(x) + x² · cos(x)

Apply the product rule: u'v + uv'

Practice Problems

Try These:

d/dx[x³ · cos(x)]
d/dx[(x+1) · (x-2)]
d/dx[e^x · ln(x)]
d/dx[√x · (x²+1)]

Answers:

3x² cos(x) - x³ sin(x)
2x - 1
e^x ln(x) + e^x/x
(5x² + 1)/(2√x)

Need Help with Product Rule Calculations?

Use our free derivative calculator with step-by-step solutions

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Related Derivative Rules

Power Rule

For x^n functions

Chain Rule

For composite functions

Quotient Rule

For division of functions