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Derivatives · Inverse Trigonometric

Derivative of arccos(x)

ddx[arccos⁡x]=−11−x2\frac{d}{dx}[\arccos x] = -\frac{1}{\sqrt{1 - x^2}}

The derivative of the inverse cosine function. Note the negative sign compared to arcsin.

Conditions. -1 < x < 1.

Worked examples

Find d/dx[arccos(x²)].
  1. Chain rule: -1/√(1-x⁴) · 2x = -2x/√(1-x⁴)

Answer: -2x/√(1 - x⁴)

Verify that d/dx[arcsin x + arccos x] = 0.
  1. d/dx[arcsin x] = 1/√(1-x²)
  2. d/dx[arccos x] = -1/√(1-x²)
  3. Sum = 0 ✓ (confirming arcsin x + arccos x = π/2)

Answer: 0

Related formulas

Derivative of arcsin(x)

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