Inverse Trigonometric Derivatives

For a differentiable inner function \(u=u(x)\), the chain rule multiplies each standard inverse-trigonometric derivative by \(u'(x)\).

\(\frac{\mathrm d}{\mathrm dx}\arcsin u=\frac{u'}{\sqrt{1-u^2}}\), valid where \(|u|<1\).

\(\frac{\mathrm d}{\mathrm dx}\arccos u=-\frac{u'}{\sqrt{1-u^2}}\), valid where \(|u|<1\).

\(\frac{\mathrm d}{\mathrm dx}\arctan u=\frac{u'}{1+u^2}\), valid wherever \(u\) is differentiable.

Check the sign before simplifying: the derivative of \(\arccos u\) carries a minus sign. For \(u=2x-1\), the derivative is \(-2/\sqrt{1-(2x-1)^2}\) on \(0<x<1\).