设函数 $z=z(x, y)$ 由 $z+\ln z-\int_y^x e^{-t^2} d t=0$ 确定，则 $\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}=$
A. $\frac{z}{z+1}\left(e^{-x^2}-e^{-y^2}\right)$
B. $\frac{z}{z+1}\left(e^{-x^2}+e^{-y^2}\right)$
C. $-\frac{z}{z+1}\left(e^{-x^2}-e^{-y^2}\right)$
D. $-\frac{z}{z+1}\left(e^{-x^2}+e^{-y^2}\right)$