170. Find the derivatives of the following functions:
1) $y=5 \arcsin k x+3 \arccos k x$; 2) $y=\arcsin \frac{a}{x}-\operatorname{arcctg} \frac{x}{a}$;
2) $r=\operatorname{arctg} \frac{m}{\varphi}+\operatorname{arcctg}(m \operatorname{ctg} \varphi)$; $r^{\prime}(0)$ ?, $r^{\prime}(\pi)$ ?
Solution. 1) Using formulas 12 and 13, we find

$$
\begin{gathered}
y^{\prime}=5 \frac{(k x)^{\prime}}{\sqrt{1-(k x)^{2}}}+3\left[-\frac{(k x)^{\prime}}{\sqrt{1-(k x)^{2}}}\right]=\frac{5 k}{\sqrt{1-k^{2} x^{2}}}- \\
-\frac{3 k}{\sqrt{1-k^{2} x^{2}}}=\frac{2 k}{\sqrt{1-k^{2} x^{2}}}
\end{gathered}
$$

2) Using formulas 12 and 15, we have

$$
\begin{gathered}
y^{\prime}=\frac{\left(\frac{a}{x}\right)^{\prime}}{\sqrt{1-\frac{a^{2}}{x^{2}}}}-\left[-\frac{\left(\frac{x}{a}\right)^{\prime}}{1+\frac{x^{2}}{a^{2}}}\right]= \\
=\frac{-\frac{a}{x^{2}}}{\sqrt{\frac{x^{2}-a^{2}}{x^{2}}}+\frac{\frac{-a^{2}}{x^{2}}}{\frac{a^{2}+x^{2}}{a^{2}}}=\frac{a}{a^{2}+x^{2}}-\frac{a}{|x| \sqrt{x^{2}-a^{2}}}},
\end{gathered}
$$

since $\sqrt{x^{2}}$ equals not $x$, but $|x|$ and $x \neq 0$.

3) Applying formulas 14 and 15:

$$
\begin{gathered}
r^{\prime}=\frac{\left(\frac{m}{\varphi}\right)^{\prime}}{1+\frac{m^{2}}{\varphi^{2}}}-\frac{(m \operatorname{ctg} \varphi)^{\prime}}{1+m^{2} \operatorname{ctg}^{2} \varphi}=\frac{-\frac{m}{\varphi^{2}}}{\frac{\varphi^{2}+m^{2}}{\varphi^{2}}}-\frac{-m \operatorname{cosec}^{2} \varphi}{1+m^{2} \operatorname{ctg}^{2} \varphi}= \\
=-\frac{m}{\varphi^{2}+m^{2}}+\frac{m}{\sin ^{2} \varphi+m^{2} \cos ^{2} \varphi}
\end{gathered}
$$

Substituting the given values of $\varphi$ which are 0 and $\pi$, we find

$$
r^{\prime}(0)=-\frac{1}{m}+\frac{1}{m}=0 ; \quad r^{\prime}(\pi)=-\frac{m}{\pi^{2}+m^{2}}+\frac{1}{m}=\frac{\pi^{2}}{m\left(\pi^{2}+m^{2}\right)}
$$

Find the derivatives of the following functions: