6.32. Find the first differential of the function $y(x)=$ $=\operatorname{arctg}(1 / x)$ and calculate its value at $x=1, d x=$ $=\Delta x=-0.1$.
Solution.

$$
\begin{aligned}
& d y(x)=\frac{1}{1+1 / x^{2}} \cdot\left(-1 / x^{2}\right) d x=-\frac{1}{x^{2}+1} d x \\
& d y(1)=-\frac{1}{1^{2}+1} \cdot(-0.1)=0.05
\end{aligned}
$$