<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 200px;"><colgroup><col style="width: 100px;"><col style="width: 100px;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center">\(\mathrm{f}(x)\)</th><th style="text-align: center; --cell-justify: center">\(\mathrm{f}'(x)\)</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">\(\sin^{-1} x\)</td><td style="text-align: center; --cell-justify: center">\(\frac{1}{\sqrt{1-x^2}}\)</td></tr><tr><td style="text-align: center; --cell-justify: center">\(\cos^{-1} x\)</td><td style="text-align: center; --cell-justify: center">\(-\frac{1}{\sqrt{1-x^2}}\)</td></tr><tr><td style="text-align: center; --cell-justify: center">\(\tan^{-1} x\)</td><td style="text-align: center; --cell-justify: center">\(\frac{1}{1+x^2}\)</td></tr><tr><td style="text-align: center; --cell-justify: center">\(\csc x\)</td><td style="text-align: center; --cell-justify: center">\(-\csc x \cot x\)</td></tr><tr><td style="text-align: center; --cell-justify: center">\(\sec x\)</td><td style="text-align: center; --cell-justify: center">\(\sec x \tan x\)</td></tr></tbody></table></div></div>

The following standard derivatives are stated in MF27 or follow directly from the standard trigonometric, exponential and logarithmic results.

Inverse-trigonometric and reciprocal-trigonometric derivatives

Function	Derivative	Conditions
\(\sin^{-1}x\)	\(\dfrac{1}{\sqrt{1-x^2}}\)	\(\|x\|<1\)
\(\cos^{-1}x\)	\(-\dfrac{1}{\sqrt{1-x^2}}\)	\(\|x\|<1\)
\(\tan^{-1}x\)	\(\dfrac{1}{1+x^2}\)	All real \(x\)
\(\mathrm{cosec}\,x\)	\(-\mathrm{cosec}\,x\cot x\)	Where defined
\(\sec x\)	\(\sec x\tan x\)	Where defined

Derivatives in MF27

Derivatives in MF27