Chain Rule

advancedcalculusderivatives

Differentiates composite functions

The Formula

d/dx[f(g(x))] = f'(g(x))·g'(x)

Variables & Symbols

f, g

Functions

f', g'

Derivatives

When to Use This Formula

  • Composite function derivatives
  • Most complex derivatives

Worked Examples

Example 1

Problem:

d/dx[sin(x²)]

Solution: 2x·cos(x²)

Step-by-Step:

1

Outer: f(u)=sin(u), Inner: g(x)=x²

2

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

3

Result: cos(x²)·2x = 2x·cos(x²)

Common Mistakes to Avoid

  • Not identifying inner/outer functions
  • Forgetting g'(x)

Real-World Applications

1

Physics

2

Engineering

3

Advanced calculus

Practice Problems

Problem 1: d/dx[(x³+1)⁵]

Show Answer

Answer: 5(x³+1)⁴·3x²

Related Formulas

power rule derivative

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product rule

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Tags:

calculusderivativechain rule

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