$3$\displaystyle{{ $ { \sin }^{ - 1 }x $ }^{\prime }=\frac{\hspace{2}1\hspace{2}}{\hspace{2} \sqrt{ 1- {x}^{2} } \hspace{2}}}$

$3$\displaystyle{ y={ \sin }^{ - 1 }x }$ とすると，
$3$\displaystyle{ x=\sin y }$ と書きかえれる。
(⇒アークサイン参照)
$3$\displaystyle{ \frac{\hspace{2} dy \hspace{2}}{\hspace{2} dx \hspace{2}} =\frac{\hspace{2}1\hspace{2}}{\hspace{2} \text{\hspace{1}}\frac{\hspace{2} dx \hspace{2}}{\hspace{2} dy \hspace{2}}\text{\hspace{1}} \hspace{2}}}$
$3$ =\frac{\hspace{2}1\hspace{2}}{\hspace{2} \cos y \hspace{2}} $ {}^\bullet\limits{ }_\bullet {}^\bullet x=\sin y\rightarrow \frac{\hspace{2} dy \hspace{2}}{\hspace{2} dx \hspace{2}}=\cos y $$
$3$\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2} \sqrt{ 1- { \sin }^{2}y } \hspace{2}} }$
$3$\displaystyle{ $\begin{array}{lll}{}^\bullet\limits{ }_\bullet {}^\bullet - \frac{\hspace{2}2\hspace{2}}{\hspace{2}{\pi}\hspace{2}}\leq y\leq \frac{\hspace{2}2\hspace{2}}{\hspace{2}{\pi}\hspace{2}}\rightarrow \cos x\geq 0& \vspace{6}\\ \text{\hspace{5}}\text{\hspace{5}}\rightarrow \cos x=\sqrt{ 1- { \sin }^{2}y }& \vspace{6}\\ \end{array}$ }$
$3$\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2} \sqrt{ 1- {x}^{2} } \hspace{2}} }$

$3$\displaystyle{ {}_\bullet\limits{ }^\bullet {}_\bullet { $ { \sin }^{ - 1 }x $ }^{\prime }=\frac{\hspace{2}1\hspace{2}}{\hspace{2} \sqrt{ 1- {x}^{2} } \hspace{2}} }$

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