Question

1.求下列复合函数的全导数或偏导数(2) z=(x-y)^5 ,其中 x=st^2 , y=s^2

Answer 1

首先，我们有：
$$\frac{\partial z}{\partial x} = 5(x-y)^4\frac{\partial}{\partial x}(x-y) = 5(x-y)^4$$
$$\frac{\partial z}{\partial y} = 5(x-y)^4\frac{\partial}{\partial y}(x-y) = 5(y-x)^4$$
然后，我们需要计算 $x,y$ 关于 $s,t$ 的偏导数。根据链式法则，我们有：
$$\frac{\partial x}{\partial s} = \frac{\partial}{\partial s}(st^2)=t^2\frac{\partial}{\partial s}(s)=t^2$$
$$\frac{\partial x}{\partial t} = \frac{\partial}{\partial t}(st^2)=s\frac{\partial}{\partial t}(t^2)=2st$$
$$\frac{\partial y}{\partial s} = \frac{\partial}{\partial s}(s^2)=2s$$
$$\frac{\partial y}{\partial t} = 0$$
最后，根据链式法则，我们有：
$$\frac{\partial z}{\partial s} = \frac{\partial z}{\partial x}\frac{\partial x}{\partial s}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial s} = 5(st^2-s^2)^4(t^2)+5(s^2-st^2)^4(2s)$$
$$\frac{\partial z}{\partial t} = \frac{\partial z}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial t} = 5(st^2-s^2)^4(2st)$$
因此，该函数的全导数为：
$$\frac{\partial z}{\partial s} = 5(st^2-s^2)^4(t^2)+10s^2(s^2-st^2)^4$$
$$\frac{\partial z}{\partial t} = 10st(st^2-s^2)^4$$

Answer 2

全导如下∵

Answer 3

本题详细计算步骤如下图所示：