设函数 $f(u, v)$ 满足 $f\left(x+y, \frac{y}{x}\right)=x^2-y^2$ ，则 $\left.\frac{\partial f}{\partial u}\right|_{\substack{u=1 \\ v=1}}$与 $\left.\frac{\partial f}{\partial v}\right|_{\substack{u=1 \\ v=1}}$ 依次是
$\text{A.}$ $\frac{1}{2}, 0$
$\text{B.}$ $0, \frac{1}{2}$
$\text{C.}$ $-\frac{1}{2}, 0$
$\text{D.}$ $0,-\frac{1}{2}$