For a differentiable inner function u=u(x), the chain rule multiplies each standard inverse-trigonometric derivative by u'(x). (d)/(dx)arcsin u=(u')/(sqrt(1-u2)), valid where |u|<1. (d)/(dx)arccos u=-(u')/(sqrt(1-u2)), valid where |u|<1. (d)/(dx)arctan u=(u')/(1+u2), 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 - 2sqrt(1-(2x-1)2) on 0<x<1.

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 \(-\dfrac{2}{\sqrt{1-(2x-1)^2}}\) on \(0<x<1\).

Techniques of Differentiation

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Inverse Trigonometric Derivatives

Inverse Trigonometric Derivatives