## Task Condition

Find the derivative.

$$
y=\frac{(1+x) \operatorname{arctg} \sqrt{x}-\sqrt{x}}{x}
$$
## Solution

$y^{\prime}=\left(\frac{(1+x) \operatorname{arctg} \sqrt{x}-\sqrt{x}}{x}\right)^{\prime}=\left(\left(\frac{1}{x}+1\right) \operatorname{arctg} \sqrt{x}-\frac{1}{\sqrt{x}}\right)^{\prime}=$
$=-\frac{1}{x^{2}} \cdot \operatorname{arctg} \sqrt{x}+\left(\frac{1}{x}+1\right) \cdot \frac{1}{1+(\sqrt{x})^{2}} \cdot \frac{1}{2 \sqrt{x}}-\left(-\frac{1}{2}\right) \cdot \frac{1}{\sqrt{x^{3}}}=$
$=-\frac{1}{x^{2}} \cdot \operatorname{arctg} \sqrt{x}+\frac{1+x}{x} \cdot \frac{1}{1+x} \cdot \frac{1}{2 \sqrt{x}}+\frac{1}{2 x \sqrt{x}}=$

$=-\frac{1}{x^{2}} \cdot \operatorname{arctg} \sqrt{x}+\frac{1}{2 x \sqrt{x}}+\frac{1}{2 x \sqrt{x}}=-\frac{1}{x^{2}} \cdot \operatorname{arctg} \sqrt{x}+\frac{1}{x \sqrt{x}}$

## Problem Kuznetsov Differentiation 10-18