# 2次の偏微分

## ■問題

$z={tan}^{-1}\text{\hspace{0.17em}}\frac{y}{x}$

## ■答

$\frac{{\partial }^{2}z}{\partial {x}^{2}}$ = $\frac{2xy}{{\left({x}^{2}+{y}^{2}\right)}^{2}}$$\frac{\partial z}{\partial y}$=$\frac{-2xy}{{\left({x}^{2}+{y}^{2}\right)}^{2}}$$\frac{{\partial }^{2}z}{\partial x\partial y}$=$\frac{-{x}^{2}+{y}^{2}}{{\left({x}^{2}+{y}^{2}\right)}^{2}}$

## ■ヒント

2次偏導関数$\frac{{\partial }^{2}z}{\partial {x}^{2}}$$\frac{{\partial }^{2}z}{\partial {y}^{2}}$$\frac{{\partial }^{2}z}{\partial y\partial x}$$\frac{{\partial }^{2}z}{\partial x\partial y}$ の4つを求める．

$\frac{\partial z}{\partial x}$$\frac{\partial z}{\partial y}$ を計算してから，それぞれを更に$x$$y$で偏微分する．

## ■解説

$z$$={tan}^{-1}\text{\hspace{0.17em}}\frac{y}{x}$

$={\mathrm{tan}}^{-1}\left(y{x}^{-1}\right)$

### ●$\frac{\partial z}{\partial x}$ の計算

$z={\mathrm{tan}}^{-1}\left(y{x}^{-1}\right)$偏導関数の定義より， $y$ を定数とみなして$x$ で微分する．

$\frac{\partial z}{\partial x}=\frac{\partial }{\partial x}{\mathrm{tan}}^{-1}\left(y{x}^{-1}\right)$

$=\frac{1}{1+{\left(y{x}^{-1}\right)}^{2}}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\left(-y{x}^{-2}\right)$

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

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

$=\frac{-y}{{x}^{2}+{y}^{2}}$

$=-y{\left({x}^{2}+{y}^{2}\right)}^{-1}$　･･････(1)

### ●$\frac{\partial z}{\partial y}$ の計算

$z={\mathrm{tan}}^{-1}\left(y{x}^{-1}\right)$偏導関数の定義より， $x$ を定数とみなして$y$ で微分する．

$\frac{\partial z}{\partial y}=\frac{\partial }{\partial y}{\mathrm{tan}}^{-1}\left(y{x}^{-1}\right)$

$=\frac{1}{1+{\left(y{x}^{-1}\right)}^{2}}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}{x}^{-1}$

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

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

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

$=x{\left({x}^{2}+{y}^{2}\right)}^{-1}$　･･････(2)

### ●$\frac{{\partial }^{2}z}{\partial {x}^{2}}$ の計算

(1)を更に， 偏導関数の定義より， $y$ を定数とみなして $x$ で微分する．

$\frac{{\partial }^{2}z}{\partial {x}^{2}}$$=\frac{\partial }{\partial x}\left(\frac{\partial z}{\partial x}\right)$

$=\frac{\partial }{\partial x}\left\{-y{\left({x}^{2}+{y}^{2}\right)}^{-1}\right\}$

$=y{\left({x}^{2}+{y}^{2}\right)}^{-2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2x$

$=\frac{y}{{\left({x}^{2}+{y}^{2}\right)}^{2}}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2x$

$=\frac{2xy}{{\left({x}^{2}+{y}^{2}\right)}^{2}}$

### ●$\frac{{\partial }^{2}z}{\partial {y}^{2}}$ の計算

(2)を更に， 偏導関数の定義より， $x$ を定数とみなして $y$ で微分する．

$\frac{{\partial }^{2}z}{\partial {y}^{2}}$$=\frac{\partial }{\partial y}\left(\frac{\partial z}{\partial y}\right)$

$=\frac{\partial }{\partial y}\left\{x{\left({x}^{2}+{y}^{2}\right)}^{-1}\right\}$

$=-x{\left({x}^{2}+{y}^{2}\right)}^{-2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2y$

$=\frac{-x}{{\left({x}^{2}+{y}^{2}\right)}^{2}}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2y$

$=\frac{-2xy}{{\left({x}^{2}+{y}^{2}\right)}^{2}}$

### ●$\frac{{\partial }^{2}z}{\partial y\partial x}$ の計算

(1)を更に，偏導関数の定義より， $x$ を定数とみなして $y$ で微分する．

$\frac{{\partial }^{2}z}{\partial y\partial x}$$=\frac{\partial }{\partial y}\left(\frac{\partial z}{\partial x}\right)$

$=\frac{\partial }{\partial y}\left\{-y{\left({x}^{2}+{y}^{2}\right)}^{-1}\right\}$

$=\frac{\partial }{\partial y}\left(-y\right)\cdot {\left({x}^{2}+{y}^{2}\right)}^{-1}+\left(-y\right)\cdot \frac{\partial }{\partial y}{\left({x}^{2}+{y}^{2}\right)}^{-1}$

$=-{\left({x}^{2}+{y}^{2}\right)}^{-1}+\left(-y\right)\cdot \frac{\partial }{\partial y}{\left({x}^{2}+{y}^{2}\right)}^{-2}\cdot 2y$

$=-{\left({x}^{2}+{y}^{2}\right)}^{-1}+2{y}^{2}{\left({x}^{2}+{y}^{2}\right)}^{-2}$

$=\frac{-1}{{x}^{2}+{y}^{2}}+\frac{2{y}^{2}}{{\left({x}^{2}+{y}^{2}\right)}^{2}}$

$=\frac{-\left({x}^{2}+{y}^{2}\right)+2{y}^{2}}{{\left({x}^{2}+{y}^{2}\right)}^{2}}$

$=\frac{-{x}^{2}-{y}^{2}+2{y}^{2}}{{\left({x}^{2}+{y}^{2}\right)}^{2}}$

$=\frac{-{x}^{2}+{y}^{2}}{{\left({x}^{2}+{y}^{2}\right)}^{2}}$　･･････(3)

### ●$\frac{{\partial }^{2}z}{\partial x\partial y}$ の計算

(2)を更に，偏導関数の定義より， $y$ を定数とみなして $x$ で微分する．

$\frac{{\partial }^{2}z}{\partial x\partial y}$$=\frac{\partial }{\partial x}\left(\frac{\partial z}{\partial y}\right)$

$=\frac{\partial z}{\partial x}\left\{x{\left({x}^{2}+{y}^{2}\right)}^{-1}\right\}$

$={\left({x}^{2}+{y}^{2}\right)}^{-1}+x\cdot \frac{\partial }{\partial x}{\left({x}^{2}+{y}^{2}\right)}^{-1}$

$={\left({x}^{2}+{y}^{2}\right)}^{-1}+x\cdot \left(-1\right)\cdot {\left({x}^{2}+{y}^{2}\right)}^{-2}\cdot 2x$

$=\frac{{x}^{2}+{y}^{2}-2{x}^{2}}{{\left({x}^{2}+{y}^{2}\right)}^{2}}$

$=\frac{-{x}^{2}+{y}^{2}}{{\left({x}^{2}+{y}^{2}\right)}^{2}}$　･･････(4)

(3)，(4)より

$\frac{{\partial }^{2}z}{\partial y\partial x}=\frac{{\partial }^{2}z}{\partial x\partial y}$

となり，2次偏導関数は偏微分する順序には無関係であることが確かめられた．

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