学习笔记0

设函数$z=f[(xy),yg(x)]​$,其中函数$f​$具有二阶连续偏导数，函数$g(x)​$可导，且在$x=1​$出取得极值$g(1)=1​$,求$left.frac{partial^2z}{partial x partial y}right|_{x=1,y=1}​$

$dot{f(x)}代表f(x)一阶导数​$

$because z=f[(xy),yg(x)] ​$

$therefore frac{partial z}{partial x}=ydot{f_{1}}+ydot{g(x)}dot{f_2}​$

$therefore frac{partial ^2z}{partial xpartial y}=dot{f_1}+y[xddot{f1}+g(x)ddot{f{12}}]+dot{g(x)}dot{f2}+ydot{g(x)}[xddot{f{12}}+g(x)ddot{f_{22}}]​$

$because g(1)=1,dot{g(1)}=0​$

$therefore left.frac{partial^2 z}{partial x partial y}right|_{x=1,y=1}=dot{f1}(1,1)+ddot{f{11}}(1,1)+ddot{f_{12}}(1,1)$

$p(x)=begin{cases} p,& x=1 1-p,& x=0end{cases}$