[m]\frac{\partial z }{\partial y}=(ln(x+e^{-y}))`=\frac{1}{x+e^{-y}}\cdot(x+e^{-y})`_{y}=\frac{1}{x+e^{-y}}\cdot e^{-y}\cdot (-y)`=\frac{-e^{-y}}{x+e^{-y}}[/m]
[m]\frac{\partial^2 z }{\partial x^2}=\frac{\partial (\frac{\partial z }{\partial x}) }{\partial x}=(\frac{1}{x+e^{-y}})`_{x}=-\frac{1}{(x+e^{-y})^2}\cdot(x+e^{-y})`_{x}= -\frac{1}{(x+e^{-y})^2}[/m]
[m]\frac{\partial^2 z }{\partial x\partial y}=\frac{\partial (\frac{\partial z }{\partial x}) }{\partial y}=(\frac{1}{x+e^{-y}})`_{y}=-\frac{1}{(x+e^{-y})^2}\cdot(x+e^{-y})`_{y}= \frac{e^{-y}}{(x+e^{-y})^2}[/m]
[m]\frac{\partial^2 z }{\partial y^2}=\frac{\partial (\frac{\partial z }{\partial y}) }{\partial y}=(\frac{-e^{-y}}{x+e^{-y}})`_{y}=[/m]
Применяем формулу производная дроби:
[m](\frac{-e^{-y}}{x+e^{-y}})`_{y}=\frac{(-e^{-y})`_{y}\cdot (x+e^{-y})-(-e^{-y})\cdot(x+e^{-y})`_{y}}{(x+e^{-y})^2}=[/m]
[m]=\frac{e^{-y}\cdot (x+e^{-y})-(-e^{-y})\cdot(-e^{-y})}{(x+e^{-y})^2}=[/m]
[m]=\frac{e^{-y}\cdot x+(e^{-y})^2-(e^{-y})^2}{(x+e^{-y})^2}=[/m]
[m]=\frac{x\cdot e^{-y}}{(x+e^{-y})^2}[/m]