第6章 6.4

(1)

$z = x^3 + 3x^2 y + y^3$

$z_x = 3x^2 + 6xy \qquad z_y = 3x^2 + 3y^2$

$z_{xx} = 6x + 6y \qquad z_{xy} = 6x \qquad z_{yy} = 6y$

(2)

$z = (x^2+y)^3$

$z_x = 3(x^2+y)^2 \cdot 2x = 6x(x^2+y)^2 \qquad z_y = 3(x^2+y)^2$

$z_{xx} = 6(x^2+y)^2 + 6x \cdot 2(x^2+y) \cdot 2x = (x^2+y)(6x^2+6y+24x^2) = (x^2+y)(30x^2+6y)$

$z_{xy} = 12x(x^2+y) \qquad z_{yy} = 6(x^2+y)$

(3)

$z = \frac{x}{x+y} = 1 - \frac{y}{x+y}$

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

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

$z_{yy} = \frac{2x}{(x+y)^3}$

(4)

$z = \log x^2 y$

$z_x = \frac{2xy}{x^2 y} = \frac{2}{x} \qquad z_y = \frac{x^2}{x^2 y} = \frac{1}{y}$

$z_{xx} = -\frac{2}{x^2} \qquad z_{xy} = 0 \qquad z_{yy} = -\frac{1}{y^2}$

(5)

$z = e^{2x} \cos 3y$

$z_x = 2e^{2x} \cos 3y \qquad z_y = -3e^{2x} \sin 3y$

$z_{xx} = 4e^{2x} \cos 3y \qquad z_{xy} = -6e^{2x} \sin 3y \qquad z_{yy} = -9e^{2x} \cos 3y$

(6)

$z = \sin x \cdot \cos y$

$z_x = \cos x \cdot \cos y \qquad z_y = -\sin x \cdot \sin y$

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

(7)

$z = x^y \qquad (x > 0)$

$z_x = yx^{y-1} \qquad z_y = x^y \log x$

$z_{xx} = y(y-1)x^{y-2} \qquad z_{xy} = x^{y-1} + yx^{y-1}\log x \qquad z_{yy} = x^y(\log x)^2$

(8)

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

$z_x = \frac{-\dfrac{2y}{4x^2}}{1+\left(\dfrac{y}{2x}\right)^2} = \frac{-2y}{4x^2+y^2} \qquad z_y = \frac{2x}{y^2+4x^2}$

$z_{xx} = \frac{16xy}{(4x^2+y^2)^2} \qquad z_{xy} = \frac{-2(4x^2+y^2)+2y \cdot 2y}{(4x^2+y^2)^2} = \frac{-8x^2+2y^2}{(4x^2+y^2)^2} \qquad z_{yy} = \frac{-4xy}{(y^2+4x^2)^2}$