Example 5. Find the derivative of the function $y=\arcsin \frac{1}{x}$.
Solution. Using formula (2.16'), we get

$$
\begin{gathered}
\left(\arcsin \frac{1}{x}\right)^{\prime}=\frac{1}{\sqrt{1-\left(\frac{1}{x}\right)^{2}}}\left(\frac{1}{x}\right)^{\prime}=\frac{1}{\sqrt{\frac{x^{2}-1}{x^{2}}}}\left(-\frac{1}{x^{2}}\right)= \\
=-\frac{1}{|x| \sqrt{x^{2}-1}}
\end{gathered}
$$