Question

Найти производную
Срочно
$xarcsin\sqrt{ \frac{x}{x + 1} } - \sqrt{x} + arctg \sqrt{x}$
​

Answer 1

$y=x\cdot arcsin\sqrt{\frac{x}{x+1}}-\sqrt{x}+arctg\sqrt{x}\\\\y'=arcsin\sqrt{\frac{x}{x+1}}+x\cdot \frac{1}{\sqrt{1-\frac{x}{x+1}}}\cdot \frac{1}{2}\cdot \sqrt{\frac{x+1}{x}}\cdot \frac{x+1-x}{(x+1)^2}-\frac{1}{2\sqrt{x}}+\frac{1}{1+x}\cdot \frac{1}{2\sqrt{x}}=\\\\=arcsin\sqrt{\frac{x}{x+1}}+\frac{x\sqrt{x+1}}{2(x+1)^2}\cdot \sqrt{\frac{x+1}{x}}-\frac{1}{2\sqrt{x}}+\frac{1}{2\sqrt{x}\cdot (1+x)}=\\\\=arcsin\sqrt{\frac{x}{x+1}}+\frac{\sqrt{x}}{2(x+1)}-\frac{1}{2\sqrt{x}}+\frac{1}{2\sqrt{x}\cdot (1+x)}$