2次の偏微分

■問題

次の関数の第2次偏導関数を求めよ．

$3$\displaystyle{ z=\sqrt{ \frac{\hspace{2}x\hspace{2}}{\hspace{2}y\hspace{2}} } }$

■答

$3$\displaystyle{ \frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}{x}^{2} \hspace{2}}=- \frac{\hspace{2}1\hspace{2}}{\hspace{2} 4x\sqrt{ xy } \hspace{2}} }$ ， $3$\displaystyle{ \frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}{y}^{2} \hspace{2}}=\frac{\hspace{2} 3\sqrt{x} \hspace{2}}{\hspace{2} 4{y}^{2}\sqrt{y} \hspace{2}} }$ ， $3$\displaystyle{ \frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}x{\partial}y \hspace{2}}=- \frac{\hspace{2}1\hspace{2}}{\hspace{2} 4y\sqrt{ xy } \hspace{2}} }$

■ヒント

2次偏導関数 $3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}{x}^{2} \hspace{2}}}$ ， $3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}{y}^{2} \hspace{2}}}$ ， $3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}y{\partial}x \hspace{2}}}$ ， $3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}x{\partial}y \hspace{2}}}$ の4つを求める．

$3$\displaystyle{\frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}}$ ， $3$\displaystyle{\frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}y \hspace{2}}}$ を計算してから，それぞれを更に $3$x$ ， $3$y$ で偏微分する．

■解説

与式を変形すると，

$3$\displaystyle{ z=\sqrt{ \frac{\hspace{2}x\hspace{2}}{\hspace{2}y\hspace{2}} } }$

平方根を累乗根の指数に変形する．指数が有理数の場合も参照．

$3$\displaystyle{ ={x}^{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} }{y}^{ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} } }$

● $3$\displaystyle{\frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}}$ の計算

$3$\displaystyle{z={x}^{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} }{y}^{ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} }}$ を偏導関数の定義より， $3$\displaystyle{y}$ を定数とみなして $3$\displaystyle{x}$ で微分する．

$3$\displaystyle{\frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}=\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\left({ {x}^{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} }{y}^{ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} } }\right)}$ $3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\cdot {x}^{ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} }{y}^{ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} }}$ $3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2} 2\sqrt{ xy } \hspace{2}}}$ 　･･････(1)

● $3$\displaystyle{\frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}y \hspace{2}}}$ の計算

$3$\displaystyle{z={x}^{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} }{y}^{ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} }}$ を偏導関数の定義より， $3$\displaystyle{x}$ を定数とみなして $3$\displaystyle{y}$ で微分する．

$3$\displaystyle{\frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}y \hspace{2}}=\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}y \hspace{2}}\left({ {x}^{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} }{y}^{ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} } }\right)}$ $3$\displaystyle{={x}^{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} }\cdot \left({ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} }\right){y}^{ - \frac{\hspace{2}3\hspace{2}}{\hspace{2}2\hspace{2}} }}$ $3$\displaystyle{=- \frac{\hspace{2} \sqrt{x} \hspace{2}}{\hspace{2} 2y\sqrt{y} \hspace{2}}}$ 　･･････(2)

● $3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}{x}^{2} \hspace{2}}}$ の計算

(1)を更に，偏導関数の定義より， $3$\displaystyle{y}$ を定数とみなして $3$\displaystyle{x}$ で微分する．

$3$\displaystyle{ \frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}{x}^{2} \hspace{2}}=\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\cdot $ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} $ {x}^{ - \frac{\hspace{2}3\hspace{2}}{\hspace{2}2\hspace{2}} }{y}^{ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} } }$ $3$\displaystyle{ =- \frac{\hspace{2}1\hspace{2}}{\hspace{2} 4x\sqrt{ xy } \hspace{2}} }$

● $3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}{y}^{2} \hspace{2}}}$ の計算

(2)を更に，偏導関数の定義より， $3$\displaystyle{x}$ を定数とみなして $3$\displaystyle{y}$ で微分する．

$3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}{y}^{2} \hspace{2}}}$ $3$\displaystyle{=\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}y \hspace{2}} $ \frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}y \hspace{2}} $ }$ $3$\displaystyle{ =\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}y \hspace{2}} $ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{x}^{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} }{y}^{ - \frac{\hspace{2}3\hspace{2}}{\hspace{2}2\hspace{2}} } $ }$ $3$\displaystyle{ =- \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\cdot $ - \frac{\hspace{2}3\hspace{2}}{\hspace{2}2\hspace{2}} $ {x}^{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} }{y}^{ - \frac{\hspace{2}5\hspace{2}}{\hspace{2}2\hspace{2}} } }$ $3$\displaystyle{ =\frac{\hspace{2} 3\sqrt{x} \hspace{2}}{\hspace{2} 4{y}^{2}\sqrt{y} \hspace{2}} }$

● $3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}y{\partial}x \hspace{2}}}$ の計算

(1)を更に，偏導関数の定義より， $3$\displaystyle{x}$ を定数とみなして $3$\displaystyle{y}$ で微分する．

$3$\displaystyle{ \frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}y{\partial}x \hspace{2}} }$ $3$\displaystyle{=\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}y \hspace{2}} $ \frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} $ }$

$3$\displaystyle{ =\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}y \hspace{2}} $ \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{x}^{ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} }{y}^{ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} } $ }$

$3$\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\cdot $ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} $ {x}^{ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} }{y}^{ - \frac{\hspace{2}3\hspace{2}}{\hspace{2}2\hspace{2}} } }$

$3$\displaystyle{ =- \frac{\hspace{2}1\hspace{2}}{\hspace{2} 4y\sqrt{ xy } \hspace{2}} }$ 　･･････(3)

● $3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}x{\partial}y \hspace{2}}}$ の計算

(2)を更に，偏導関数の定義より， $3$\displaystyle{y}$ を定数とみなして $3$\displaystyle{x}$ で微分する．

$3$\displaystyle{ \frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}x{\partial}y \hspace{2}}=\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}} $ \frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}y \hspace{2}} $ }$

$3$\displaystyle{ =\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}} $ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{x}^{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} }{y}^{ - \frac{\hspace{2}3\hspace{2}}{\hspace{2}2\hspace{2}} } $ }$

$3$\displaystyle{ =- \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\cdot \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{x}^{ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} }{y}^{ - \frac{\hspace{2}3\hspace{2}}{\hspace{2}2\hspace{2}} } }$

$3$\displaystyle{ =- \frac{\hspace{2}1\hspace{2}}{\hspace{2} 4y\sqrt{ xy } \hspace{2}} }$ 　･･････(4)

(3)，(4)より

$3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}y{\partial}x \hspace{2}}=\frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}x{\partial}y \hspace{2}}}$

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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最終更新日： 2023年8月31日

2次の偏微分

■問題

■答

■ヒント

■解説

● の計算

● の計算

● の計算

● の計算

● の計算

● の計算

● $3$\displaystyle{\frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}}$ の計算

● $3$\displaystyle{\frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}y \hspace{2}}}$ の計算

● $3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}{x}^{2} \hspace{2}}}$ の計算

● $3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}{y}^{2} \hspace{2}}}$ の計算

● $3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}y{\partial}x \hspace{2}}}$ の計算

● $3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}x{\partial}y \hspace{2}}}$ の計算