CalcRef

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.

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)

Find d/dx[(3x+1)⁵].
  1. Outer: u⁵, derivative: 5u⁴. Inner: u = 3x+1, derivative: 3
  2. 5(3x+1)⁴ · 3 = 15(3x+1)⁴

Answer: 15(3x+1)⁴

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

Answer: e^(sin x) · cos x

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