第6章 6.16

（証明）

$x, y$ を任意の定数と考えて $t$ で微分すると

$xf_{x(tx,ty)} + yf_{y(tx,ty)} = nt^{n-1}f_{(x,y)}$

ここで $t=1$ とおくと

$x\frac{\partial f}{\partial x} + y\frac{\partial f}{\partial y} = nf$

(1)

$z = \frac{xy}{x+y}$

$z_x = \frac{y(x+y)-xy}{(x+y)^2} \qquad z_y = \frac{x(x+y)-xy}{(x+y)^2}$

$xz_x + yz_y = \frac{xy(x+y)-x^2y}{(x+y)^2} + \frac{xy(x+y)-xy^2}{(x+y)^2} = \frac{xy(2x+2y-x-y)}{(x+y)^2} = \frac{xy}{x+y} = z$

(2)

$z = \frac{1}{x^2+y^2}$

$z_x = \frac{-2x}{(x^2+y^2)^2} \qquad z_y = \frac{-2y}{(x^2+y^2)^2}$

$xz_x + yz_y = \frac{-2x^2-2y^2}{(x^2+y^2)^2} = \frac{-2}{x^2+y^2} = -2z$