## Task Condition

Find the differential $d y$.

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

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

## Problem Kuznetsov Differentiation 4-22