隐函数求偏导

hbghlyj

MSE
Consider the equation$$x^y+y^z+z^x=k$$defined for $x,y$ and $z$ all positive and $k$ is a positive constant.
Find $∂z/∂x$

hbghlyj

两边对$x$求偏导吗？把$y$看作常数
\[0=\frac∂{∂x}(x^y + y^{z(x, y)} + z(x, y)^x) =x^{y - 1}y+y^z(\log y)\frac{∂z}{∂x}+ z^x\left(\frac xz\frac{∂z}{∂x} + \log z\right)\]
解出
\[\frac{∂z}{∂x}=-\frac{x^{y - 1}y+z^x \log z}{z^{x-1}x+y^z \log y}\]