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Derivatives · Basic Rules

Chain Rule

ddx[f(g(x))]=f′(g(x))⋅g′(x)\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)

For composite functions: differentiate the outer function evaluated at the inner function, then multiply by the derivative of the inner function.

  • 3 worked examples
  • 24 in Derivatives

Worked examples

Find d/dx[sin(x²)].
  1. Outer: sin(u), derivative: cos(u). Inner: u = x², derivative: 2x
  2. cos(x²) · 2x = 2x cos(x²)

Answer: 2x cos(x²)

2 more worked examplesPremium

Find d/dx[(3x+1)⁵].

Step-by-step solution locked.

Find d/dx[e^(sin x)].

Step-by-step solution locked.

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Common questions

What is the Chain Rule?

For composite functions: differentiate the outer function evaluated at the inner function, then multiply by the derivative of the inner function. It is one of the derivatives formulas in the CalcRef reference.

How do you use the Chain Rule?

Worked example. Find d/dx[sin(x²)]. Outer: sin(u), derivative: cos(u). Inner: u = x², derivative: 2x. cos(x²) · 2x = 2x cos(x²). Answer: 2x cos(x²).

Practice

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