Product Rule

intermediatecalculusderivatives

Differentiates product of two functions

The Formula

d/dx[f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x)

Variables & Symbols

f(x), g(x)

Functions of x

f'(x), g'(x)

Derivatives of functions

When to Use This Formula

  • Derivative of products
  • Complex function differentiation

Worked Examples

Example 1

Problem:

d/dx[x²·sin(x)]

Solution: 2x·sin(x) + x²·cos(x)

Step-by-Step:

1

Let f(x)=x², g(x)=sin(x)

2

f'(x)=2x, g'(x)=cos(x)

3

Apply: 2x·sin(x) + x²·cos(x)

Common Mistakes to Avoid

  • Forgetting one of the terms
  • Order confusion

Real-World Applications

1

Physics - related rates

2

Economics

3

Engineering

Practice Problems

Problem 1: d/dx[x³·eˣ]

Show Answer

Answer: 3x²·eˣ + x³·eˣ

Related Formulas

power rule derivative

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chain rule

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quotient rule

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Tags:

calculusderivativeproduct rule

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