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}$ $3$\displaystyle{=\sqrt{ 2x- 3y }}$

■答え

$3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}{x}^{2} \hspace{2}}=- \frac{\hspace{2} \sqrt{ 2x- 3y } \hspace{2}}{\hspace{2} { $ 2x- 3y $ }^{2} \hspace{2}}}$ ， $3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}{y}^{2} \hspace{2}}=- \frac{\hspace{2} 9\sqrt{ 2x- 3y } \hspace{2}}{\hspace{2} 4{ $ 2x- 3y $ }^{2} \hspace{2}}}$ ， $3$\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}}=\frac{\hspace{2} 3\sqrt{ 2x- 3y } \hspace{2}}{\hspace{2} 2{ $ 2x- 3y $ }^{2} \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}$ $3$\displaystyle{=\sqrt{ 2x- 3y }}$

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

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

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

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

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

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

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

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

$3$\displaystyle{=- \frac{\hspace{2}9\hspace{2}}{\hspace{2}4\hspace{2}}\text{\hspace{1} }\cdot \text{\hspace{1} }\frac{\hspace{2}1\hspace{2}}{\hspace{2} $ 2x- 3y $ \sqrt{ 2x- 3y } \hspace{2}}}$

$3$\displaystyle{=- \frac{\hspace{2}9\hspace{2}}{\hspace{2}4\hspace{2}}\text{\hspace{1} }\cdot \text{\hspace{1} }\frac{\hspace{2} \sqrt{ 2x- 3y } \hspace{2}}{\hspace{2} { $ 2x- 3y $ }^{2} \hspace{2}}}$

$3$\displaystyle{=- \frac{\hspace{2} 9\sqrt{ 2x- 3y } \hspace{2}}{\hspace{2} 4{ $ 2x- 3y $ }^{2} \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}} \{ { $ 2x- 3y $ }^{ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} } \} }$

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

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

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

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

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

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

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

$3$\displaystyle{=\frac{\hspace{2} 3\sqrt{ 2x- 3y } \hspace{2}}{\hspace{2} 2{ $ 2x- 3y $ }^{2} \hspace{2}}}$ 　･･････(3)

(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30日

2次の偏微分

■問題

■答え

■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