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6.2 偏導関数

1.

(a) $x$での偏微分は$x$以外の変数をすべて定数と見なして微分すればよいので，

$\displaystyle f_{x}(x,y) = \frac{\partial (3x^{2})}{\partial x} - y\frac{\partial x}{\partial x} = 6x -y$

$y$での偏微分は$y$以外の変数をすべて定数と見なして微分すればよいので，

$\displaystyle f_{y}(x,y) = -x\frac{\partial y}{\partial y} + \frac{\partial y}{\partial y} = -x + 1$

(b) $x$での偏微分は$x$以外の変数をすべて定数と見なして微分すればよいので，

$\displaystyle f_{x}(x,y) = e^{-y}\frac{\partial (x^{2})}{\partial x} = 2xe^{-y}$

$y$での偏微分は$y$以外の変数をすべて定数と見なして微分すればよいので，

$\displaystyle f_{y}(x,y) = x^{2}\frac{\partial (e^{-y})}{\partial y} = -x^{2}e^{-y}$

(c) $x^{2}+y^{2}$を$t$とおくと

$\displaystyle f_{x}(x,y)$ $\displaystyle =$ $\displaystyle \frac{\partial (\sqrt{t})}{\partial x} = \frac{\partial (\sqrt{t})}{\partial t}\frac{\partial t}{\partial x}$  
  $\displaystyle =$ $\displaystyle \frac{1}{2\sqrt{t}}\cdot 2x = \frac{2x}{2\sqrt{x^{2}+y^{2}}} = \frac{x}{\sqrt{x^{2} + y^{2}}}$  


$\displaystyle f_{y}(x,y)$ $\displaystyle =$ $\displaystyle \frac{\partial (\sqrt{t})}{\partial y} = \frac{\partial (\sqrt{t})}{\partial t}\frac{\partial t}{\partial y}$  
  $\displaystyle =$ $\displaystyle \frac{1}{2\sqrt{t}}\cdot 2y = \frac{2y}{2\sqrt{x^{2}+y^{2}}} = \frac{y}{\sqrt{x^{2} + y^{2}}}$  

(d) $x$での偏微分は$x$以外の変数をすべて定数と見なして微分すればよいので，

$\displaystyle f_{x}(x,y) = \sin{y}\frac{\partial x}{\partial x} = \sin{y}$

$y$での偏微分は$y$以外の変数をすべて定数と見なして微分すればよいので，

$\displaystyle f_{y}(x,y) = x\frac{\partial (\sin{y})}{\partial y} = x\sin{y}$

(e) 商の微分法を用いると，

$\displaystyle f_{x}(x,y)$ $\displaystyle =$ $\displaystyle \frac{\frac{\partial{(x-y)}}{\partial x}(x+y) - (x-y)\frac{\partial (x+y)}{\partial x}}{(x+y)^{2}}$  
  $\displaystyle =$ $\displaystyle \frac{x+y-(x-y)}{(x+y)^{2}} = \frac{2y}{(x+y)^{2}}$  


$\displaystyle f_{y}(x,y)$ $\displaystyle =$ $\displaystyle \frac{\frac{\partial{(x-y)}}{\partial y}(x+y) - (x-y)\frac{\partial (x+y)}{\partial y}}{(x+y)^{2}}$  
  $\displaystyle =$ $\displaystyle \frac{-(x+y)-(x-y)}{(x+y)^{2}} = \frac{-2x}{(x+y)^{2}}$  

2.

(a)

$\displaystyle f_{x}$ $\displaystyle =$ $\displaystyle \frac{\partial (ax^{2})}{\partial x} + 2by \frac{\partial x}{\partial x} = 2ax + 2by$  
$\displaystyle f_{y}$ $\displaystyle =$ $\displaystyle 2bx\frac{\partial y}{\partial y} + \frac{\partial (cy^{2})}{\partial y} = 2bx + 2cy$  
$\displaystyle f_{xx}$ $\displaystyle =$ $\displaystyle \frac{\partial (2ax)}{\partial x} = 2a$  
$\displaystyle f_{xy}$ $\displaystyle =$ $\displaystyle \frac{\partial (2by)}{\partial y} = 2b$  
$\displaystyle f_{yy}$ $\displaystyle =$ $\displaystyle \frac{\partial (2cy)}{\partial y} = 2c$  

(b)

$\displaystyle f_{x}$ $\displaystyle =$ $\displaystyle 2(x + y^{2} + z^{3})\frac{\partial (x+y^{2}+z^{3})}{\partial x} = 2(x + y^{2} + z^{3})$  
$\displaystyle f_{y}$ $\displaystyle =$ $\displaystyle 2(x + y^{2} + z^{3})\frac{\partial (x+y^{2}+z^{3})}{\partial y} = 4y(x + y^{2} + z^{3})$  
$\displaystyle f_{z}$ $\displaystyle =$ $\displaystyle 2(x + y^{2} + z^{3})\frac{\partial (x+y^{2}+z^{3})}{\partial z} = 6z^{2}(x + y^{2} + z^{3})$  
$\displaystyle f_{xx}$ $\displaystyle =$ $\displaystyle \frac{\partial (2x)}{\partial x} = 2$  
$\displaystyle f_{xy}$ $\displaystyle =$ $\displaystyle \frac{\partial (2y^{2})}{\partial y} = 4y$  
$\displaystyle f_{xz}$ $\displaystyle =$ $\displaystyle \frac{\partial (2z^{3})}{\partial z} = 6z^{2}$  
$\displaystyle f_{yy}$ $\displaystyle =$ $\displaystyle \frac{\partial (4y)}{\partial y}(x + y^{2} + z^{3}) + 4y\frac{\partial (y^{2})}{\partial y} = 4(x+ y^{2} + z^{3}) + 8y^{2}$  
$\displaystyle f_{yz}$ $\displaystyle =$ $\displaystyle \frac{\partial (4yz^{3})}{\partial z} = 12yz^{2}$  
$\displaystyle f_{zz}$ $\displaystyle =$ $\displaystyle \frac{\partial (6z^{2})}{\partial z}(x + y^{2} + z^{3}) + 6z^{2}\frac{\partial (z^{3})}{\partial z} = 12z(x + y^{2} + z^{3}) + 18z^{4}$  

(c)

$\displaystyle f_{x}$ $\displaystyle =$ $\displaystyle \frac{\partial (\sin{t})}{\partial x} = \frac{\partial (\sin{t})}... ...artial (\sin{t})}{\partial t}\frac{\partial (3x-2y)}{\partial x} = 3\cos(3x-2y)$  
$\displaystyle f_{y}$ $\displaystyle =$ $\displaystyle \frac{\partial (\sin{t})}{\partial y} = \frac{\partial (\sin{t})}... ...rtial (\sin{t})}{\partial t}\frac{\partial (3x-2y)}{\partial y} = -2\cos(3x-2y)$  
$\displaystyle f_{xx}$ $\displaystyle =$ $\displaystyle \frac{\partial (3\cos(3x-2y))}{\partial x} = 3\frac{\partial (\cos(3x-2y))}{\partial x} = -9\sin(3x-2y)$  
$\displaystyle f_{xy}$ $\displaystyle =$ $\displaystyle \frac{\partial (3\cos(3x-2y))}{\partial y} = 3\frac{\partial (\cos(3x-2y))}{\partial y} = 6\sin(3x-2y)$  
$\displaystyle f_{yy}$ $\displaystyle =$ $\displaystyle -2\frac{\partial (\cos(3x-2y))}{\partial y} = -4\sin(3x-2y)$  

(d)

$\displaystyle f_{x}$ $\displaystyle =$ $\displaystyle \frac{\partial (xe^{2y})}{\partial x} = e^{2y}\frac{\partial x}{\partial x} = e^{2y}$  
$\displaystyle f_{y}$ $\displaystyle =$ $\displaystyle \frac{\partial (xe^{2y})}{\partial y} = x\frac{\partial (e^{2y})}{\partial y} = 2xe^{2y}$  
$\displaystyle f_{xx}$ $\displaystyle =$ $\displaystyle \frac{\partial (e^{2y})}{\partial x} = 0$  
$\displaystyle f_{xy}$ $\displaystyle =$ $\displaystyle \frac{\partial (e^{2y})}{\partial y} = 2e^{2y}$  
$\displaystyle f_{yy}$ $\displaystyle =$ $\displaystyle 2x\frac{\partial (e^{2y})}{\partial y} = 4xe^{2y}$  

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Next: 6.3 関数の極限 Up: 確認問題詳解 Previous: 6.1 関数の定義   索引