## Problem Statement

Find the differential $d y$.

$$
y=\left(\sqrt{x-1}-\frac{1}{2}\right) e^{2 \sqrt{x-1}}
$$
## Solution

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

## Problem Kuznetsov Differentiation 4-26