## Task Condition

Find the derivative.
$y=(2+3 x) \sqrt{x-1}-\frac{3}{2} \operatorname{arctg} \sqrt{x-1}$
## Solution

$$
\begin{aligned}
& y^{\prime}=\left((2+3 x) \sqrt{x-1}-\frac{3}{2} \operatorname{arctg} \sqrt{x-1}\right)^{\prime}= \\
& =3 \cdot \sqrt{x-1}+(2+3 x) \cdot \frac{1}{2 \sqrt{x-1}}-\frac{3}{2} \cdot \frac{1}{1+(\sqrt{x-1})^{2}} \cdot \frac{1}{2 \sqrt{x-1}}= \\
& =\frac{3(x-1)}{\sqrt{x-1}}+\frac{2+3 x}{2 \sqrt{x-1}}-\frac{3}{2} \cdot \frac{1}{1+x-1} \cdot \frac{1}{2 \sqrt{x-1}}= \\
& =\frac{12 x(x-1)}{4 x \sqrt{x-1}}+\frac{2 x(2+3 x)}{4 x \sqrt{x-1}}-\frac{3}{4 x \sqrt{x-1}}=\frac{12 x^{2}-12 x+4 x+6 x^{2}-3}{4 x \sqrt{x-1}}= \\
& =\frac{18 x^{2}-8 x-3}{4 x \sqrt{x-1}}
\end{aligned}
$$

## Problem Kuznetsov Differentiation 14-16