719. Calculate the values of the partial derivatives of the given functions at the specified values of the arguments:
1) $f(\alpha, \beta)=\cos (m \alpha-n \beta) ; \alpha=\frac{\pi}{2 m}, \beta=0$;
2) $z=\ln \left(x^{2}-y^{2}\right) ; x=2, y=-1$.
Solution. 1) Using the differentiation formulas (Chapter II), we find the partial derivatives:

$$
f_{\alpha}^{\prime}=-m \sin (m \alpha-n \beta) ; \quad f_{\beta}^{\prime}=n \sin (m \alpha-n \beta) .
$$

Setting $\alpha=\frac{\pi}{2 m}, \beta=0$, we get $f_{\alpha}^{\prime}\left(\frac{\pi}{2 m}, 0\right)=-m ; f_{\beta}^{\prime}\left(\frac{\pi}{2 m}, 0\right)=n$.

2) We find the derivatives, then compute their partial values at the specified point:

$$
z_{x}^{\prime}=\frac{2 x}{x^{2}-y^{2}} ; \quad z_{y}^{\prime}=-\frac{2 y}{x^{2}-y^{2}} ; \quad z_{x}^{\prime}(2 ;-1)=\frac{4}{3} ; \quad z_{y}^{\prime}(2 ;-1)=\frac{2}{3}
$$