多元复合函数求导
口诀：分段用乘，分叉用加，单路全导，岔路偏导。
一、1-n-1类型
$$z=f(u,v),\begin{cases}u=\phi(t) \\v=\psi(t) \end{cases} \Rightarrow z = f[\phi(t),\psi(t)]$$
$$\dfrac{dz}{dt}=\dfrac{\partial f}{\partial u}\cdot\dfrac{du}{dt}+\dfrac{\partial f}{\partial v}\cdot\dfrac{dv}{dt}$$
二、1-n-n类型
$$z=f(u,v),\begin{cases}u=\phi(x,y) \\v=\psi(x,y) \end{cases} $$
$$\dfrac{\partial z}{\partial x}=\dfrac{\partial f}{\partial u}\cdot\dfrac{\partial u}{\partial x}+\dfrac{\partial f}{\partial v}\cdot\dfrac{\partial v}{\partial x}$$
$$\dfrac{\partial z}{\partial y}=\dfrac{\partial f}{\partial u}\cdot\dfrac{\partial u}{\partial y}+\dfrac{\partial f}{\partial v}\cdot\dfrac{\partial v}{\partial y}$$
三、1-混合-n类型
$$z=f(x,v), v=\psi(x,y) $$
$$\dfrac{\partial z}{\partial x}=\dfrac{\partial f}{\partial x} +\dfrac{\partial f}{\partial v}\cdot\dfrac{\partial v}{\partial x}$$
$$\dfrac{\partial z}{\partial y}=\dfrac{\partial f}{\partial v}\cdot\dfrac{\partial v}{\partial y}$$
四、1-混合-1类型
$$z=f(u,v,t),\begin{cases}u=\phi(t) \\v=\psi(t) \end{cases} $$
$$\dfrac{dz}{dt}=\dfrac{\partial f}{\partial u}\cdot\dfrac{du}{dt} +\dfrac{\partial f}{\partial v}\cdot\dfrac{dv}{dt} +\dfrac{\partial f}{\partial t} $$
五、1-n-1类型二阶导
$$z=f(t^2,e^t)\ ,\ f(u,v)$$二阶连续可导,求二阶导数
关键 $$f1',f2'$$均为关于u，v的复合函数$$f_u'(t^2,e^t),f_v'(t^2,e^t)$$！
简化记号$$f_1'=\dfrac{\partial f}{\partial u},f_{12}''=\dfrac{\partial ^2 f}{\partial u \partial v},\cdots$$
解答：
$$\dfrac{dz}{dt}=2tf_1'+e^tf_{2}'$$
$$\dfrac{d^2z}{dt^2}=2f_1'+2t(2tf_{11}''+e^tf_{12}'') $$
$$\qquad + e^tf_2'+e^t(2tf_{21}''+e^tf_{22}'') $$
六、1-n-n类型二阶导