## Task Condition

Find the derivative.
$y=\sqrt{\frac{2}{3}} \cdot \operatorname{arctg} \frac{3 x-1}{\sqrt{6 x}}$
## Solution

$$
\begin{aligned}
& y^{\prime}=\left(\sqrt{\frac{2}{3}} \cdot \operatorname{arctg} \frac{3 x-1}{\sqrt{6 x}}\right)^{\prime}=\sqrt{\frac{2}{3}} \cdot \frac{1}{1+\left(\frac{3 x-1}{\sqrt{6 x}}\right)^{2}}\left(\frac{3 x-1}{\sqrt{6 x}}\right)^{\prime}= \\
& =\sqrt{\frac{2}{3}} \cdot \frac{6 x}{6 x+(3 x-1)^{2}} \cdot \frac{3 \sqrt{6 x}-(3 x-1) \cdot \frac{1}{2 \sqrt{6 x}} \cdot 6}{6 x}= \\
& =\sqrt{\frac{2}{3}} \cdot \frac{1}{6 x+9 x^{2}-6 x+1} \cdot \frac{3 \cdot 6 x-(3 x-1) \cdot 3}{\sqrt{6 x}}= \\
& =\frac{1}{9 x^{2}+1} \cdot \frac{18 x-9 x+3}{3 \sqrt{x}}=\frac{1}{9 x^{2}+1} \cdot \frac{3 x+1}{\sqrt{x}}=\frac{3 x+1}{\sqrt{x}\left(9 x^{2}+1\right)}
\end{aligned}
$$

## Problem Kuznetsov Differentiation 10-6