Question

Найти производную функции

Answer 1

Ответ:

 $y=2\sqrt{e^{x}}+1+ln\dfrac{\sqrt{e^{x}+1}-1}{\sqrt{e^{x}+1}+1}\\\\\\y'=2\cdot \dfrac{e^{x}}{2\sqrt{e^{x}}}+\dfrac{\sqrt{e^{x}+1}+1}{\sqrt{e^{x}+1}-1}\cdot \dfrac{\dfrac{e^{x}}{2\sqrt{e^{x}+1}}\cdot (\sqrt{e^{x}+1}+1)-(\sqrt{e^{x}+1}-1)\cdot \dfrac{e^{x}}{2\sqrt{e^{x}+1}}}{(\sqrt{e^{x}+1}+1)^2}=\\\\\\=\sqrt{e^{x}}+\dfrac{1}{\sqrt{e^{x}+1}-1}\cdot \dfrac{\dfrac{e^{x}}{2\sqrt{e^{x}+1}}\cdot \Big(\sqrt{e^{x}+1}+1-\sqrt{e^{x}+1}+1\Big)}{\sqrt{e^{x}+1}+1}=$

$=\sqrt{e^{x}}+\dfrac{e^{x}}{\sqrt{e^{x}+1}\cdot (e^{x}+1-1)}=\sqrt{e^{x}}+\dfrac{1}{\sqrt{e^{x}+1}}$