Задача 71922 найти производную y = sqrt(1+2x) -...
Условие
y = sqrt(1+2x) - ln(x+sqrt(1+2x))
математика ВУЗ
68
Решение
★
[m]y` = (\sqrt{1+2x} )`–( ln(x+\sqrt{1+2x}))`[/m]
[m]y`=\frac{1}{2\sqrt{1+2x}}\cdot (1+2x)`-\frac{1}{x+\sqrt{1+2x})}\cdot (x+\sqrt{1+2x})`[/m]
[m]y`=\frac{1}{2\sqrt{1+2x}}\cdot (2)`-\frac{1}{(x+\sqrt{1+2x})}\cdot ((x)`+(\sqrt{1+2x})`)[/m]
[m]y`=\frac{1}{\sqrt{1+2x}}-\frac{1}{(x+\sqrt{1+2x})}\cdot (1+\frac{1}{\sqrt{1+2x}})[/m]
[m]y`=\frac{1}{\sqrt{1+2x}}-\frac{1}{(x+\sqrt{1+2x})}\cdot (\frac{\sqrt{1+2x}+1}{\sqrt{1+2x}})[/m]
[m]y`=\frac{(x+\sqrt{1+2x})-(\sqrt{1+2x}+1)}{(x+\sqrt{1+2x})\sqrt{1+2x}}[/m]
[m]y`=\frac{x-1}{(x+\sqrt{1+2x})\sqrt{1+2x}}[/m]