Condition of the problem

Find the differential $d y$.

$$
y=\ln \left|x^{2}-1\right|-\frac{1}{x^{2}-1}
$$
## Solution

$$
\begin{aligned}
& d y=y^{\prime} \cdot d x=\left(\ln \left|x^{2}-1\right|-\frac{1}{x^{2}-1}\right)^{\prime} d x=\left(\frac{1}{x^{2}-1} \cdot 2 x+\frac{1}{\left(x^{2}-1\right)^{2}} \cdot 2 x\right) d x= \\
& =\frac{2 x}{x^{2}-1} \cdot\left(1+\frac{1}{x^{2}-1}\right) d x=\frac{2 x}{x^{2}-1} \cdot \frac{x^{2}}{x^{2}-1} \cdot d x=\frac{2 x^{3}}{\left(x^{2}-1\right)^{2}} \cdot d x
\end{aligned}
$$

Kuznetsov Differentiation 4-20