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Chain Rule for Derivatives

Learn how to find derivatives of composite functions with the chain rule.

The Chain Rule Formula

If f(x) = g(h(x)), then f'(x) = g'(h(x)) · h'(x)

Derivative of outside function × derivative of inside function

🔗 Think of it as:

  • • Outside function: g(u)
  • • Inside function: u = h(x)
  • • Chain rule: dg/du × du/dx
  • • Work from outside to inside

🎯 When to use:

  • • Function inside another function
  • • sin(x²), cos(3x), e^(x²)
  • • (x² + 1)⁵, √(x² + 1)
  • • ln(x² + 3x)

Step-by-Step Examples

Example 1: Power of a Function

Find: d/dx[(x² + 1)⁵]
Step 1: Identify outside: u⁵, inside: u = x² + 1
Step 2: Derivative of outside: 5u⁴
Step 3: Derivative of inside: 2x
Step 4: Chain rule: 5u⁴ × 2x
Step 5: Substitute back: 5(x² + 1)⁴ × 2x
Answer: 10x(x² + 1)⁴

Example 2: Trigonometric Composite

Find: d/dx[sin(3x)]
Step 1: Identify outside: sin(u), inside: u = 3x
Step 2: Derivative of outside: cos(u)
Step 3: Derivative of inside: 3
Step 4: Chain rule: cos(u) × 3
Step 5: Substitute back: cos(3x) × 3
Answer: 3cos(3x)

Example 3: Exponential Composite

Find: d/dx[e^(x²)]
Step 1: Identify outside: e^u, inside: u = x²
Step 2: Derivative of outside: e^u
Step 3: Derivative of inside: 2x
Step 4: Chain rule: e^u × 2x
Step 5: Substitute back: e^(x²) × 2x
Answer: 2xe^(x²)

Example 4: Multiple Chain Rule

Find: d/dx[sin(cos(x))]
Step 1: Outermost: sin(u), where u = cos(x)
Step 2: Derivative of sin(u): cos(u)
Step 3: Now find d/dx[cos(x)]: -sin(x)
Step 4: Chain rule: cos(u) × (-sin(x))
Step 5: Substitute: cos(cos(x)) × (-sin(x))
Answer: -sin(x)cos(cos(x))

🎯 Chain Rule Strategy

Step-by-Step Process:

  1. Identify the "outside" function
  2. Identify the "inside" function
  3. Take derivative of outside (leave inside alone)
  4. Multiply by derivative of inside
  5. Simplify if possible

Recognition Tips:

  • • Look for functions "nested" inside others
  • • If you can't use basic rules, try chain rule
  • • Practice identifying inside vs outside
  • • Remember: work from outside in

Practice Problems

Try These:

  1. d/dx[(x³ + 2)⁷]
  2. d/dx[cos(x²)]
  3. d/dx[ln(x² + 1)]
  4. d/dx[√(x² + 3x)]

Answers:

  1. 21x²(x³ + 2)⁶
  2. -2x sin(x²)
  3. (2x)/(x² + 1)
  4. (2x + 3)/(2√(x² + 3x))

Need Help with Chain Rule Problems?

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Related Derivative Rules

Power Rule

For x^n functions

Product Rule

For products of functions

All Rules

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