第6章 6.11

(1)

$z = x^3 - 3axy + y^3$

$z_x = 3x^2 - 3ay \qquad z_y = -3ax + 3y^2$

$z_{xx} = 6x \qquad z_{zy} = -3a \qquad z_{yy} = 6y$

(2)

$z = \sin xy$

$z_x = y\cos xy \qquad z_y = x\cos xy$

$z_{xx} = -y^2 \sin xy \qquad z_{xy} = \cos xy - xy\sin xy \qquad z_{yy} = -x^2 \sin xy$

(3)

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

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

$z_{xx} = \frac{4x}{(x-y)^3} \qquad z_{xy} = \frac{-2(x-y)^2 - 2y\cdot 2(x-y)}{(x-y)^4} = \frac{-2x-2y}{(x-y)^3} \qquad z_{yy} = \frac{4y}{(x-y)^3}$

(4)

$z = \sin^{-1}\frac{y}{x} \qquad (x>0)$

$z_x = \frac{-\dfrac{1}{x^2}}{\sqrt{\dfrac{1}{y^2}-\dfrac{1}{x^2}}} = \frac{-y}{x\sqrt{x^2-y^2}}, \qquad z_y = \frac{1}{\sqrt{x^2-y^2}}$

$z_{xx} = \frac{\dfrac{y}{x^2}\sqrt{x^2-y^2} + \dfrac{y}{x}\cdot\dfrac{x}{\sqrt{x^2-y^2}}}{(\sqrt{x^2-y^2})^2} = \frac{(2x^2-y^2)y}{x^2(x^2-y^2)^{\frac{3}{2}}}$

$z_{xy} = \frac{-x}{\sqrt{(x^2-y^2)^3}} \qquad z_{yy} = \frac{y}{\sqrt{(x^2-y^2)^3}}$