设函数 $z=z(x, y)$ 由方程 $x-a z=\mathrm{e}^{y+a z}(a$ 是非零常数）确定，则（ ）
A
$\frac{\partial z}{\partial x}-\frac{\partial z}{\partial y}=\frac{1}{a}$
B
$\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}=\frac{1}{a}$
C
$\frac{\partial z}{\partial x}-\frac{\partial z}{\partial y}=-\frac{1}{a}$
D
$\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}=-\frac{1}{a}$
E
F