合成関数の2次偏導関数の導出

$3$\displaystyle{z=f $ x,y $ }$ で $3$\displaystyle{x={\varphi} $ u,v $ ,y={\psi} $ u,v $ }$ ならば

$3$\displaystyle{ \frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}{u}^{2} \hspace{2}}=\hspace{1}{f}_{ xx }{ $ \frac{\hspace{2} {\partial}x \hspace{2}}{\hspace{2} {\partial}u \hspace{2}} $ }^{2}+2\hspace{1}{f}_{ xy }\frac{\hspace{2} {\partial}x \hspace{2}}{\hspace{2} {\partial}u \hspace{2}}\frac{\hspace{2} {\partial}y \hspace{2}}{\hspace{2} {\partial}u \hspace{2}} }$ $3$\displaystyle{ +\hspace{1}{f}_{ yy }{ $ \frac{\hspace{2} {\partial}y \hspace{2}}{\hspace{2} {\partial}u \hspace{2}} $ }^{2} }$ $3$\displaystyle{+\hspace{1}{f}_{x}\frac{\hspace{2} {{\partial}}^{2}x \hspace{2}}{\hspace{2} {\partial}{u}^{2} \hspace{2}}+\hspace{1}{f}_{y}\frac{\hspace{2} {{\partial}}^{2}y \hspace{2}}{\hspace{2} {\partial}{u}^{2} \hspace{2}}}$

もしくは，

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

■導出

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

$3$\displaystyle{=\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}u \hspace{2}} $ \hspace{1}{f}_{x}\frac{\hspace{2} {\partial}x \hspace{2}}{\hspace{2} {\partial}u \hspace{2}}+\hspace{1}{f}_{y}\frac{\hspace{2} {\partial}y \hspace{2}}{\hspace{2} {\partial}u \hspace{2}} $ }$

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

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

$3$\displaystyle{ = \{ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}u \hspace{2}} $ \frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} $ \cdot \frac{\hspace{2} {\partial}x \hspace{2}}{\hspace{2} {\partial}u \hspace{2}}+\frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\cdot \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}u \hspace{2}} $ \frac{\hspace{2} {\partial}x \hspace{2}}{\hspace{2} {\partial}u \hspace{2}} $ \} }$ $3$\displaystyle{ + \{ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}u \hspace{2}} $ \frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}y \hspace{2}} $ \cdot \frac{\hspace{2} {\partial}y \hspace{2}}{\hspace{2} {\partial}u \hspace{2}}+\frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}y \hspace{2}}\cdot \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}u \hspace{2}} $ \frac{\hspace{2} {\partial}y \hspace{2}}{\hspace{2} {\partial}u \hspace{2}} $ \} }$

$3$\displaystyle{= \{ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}u \hspace{2}} $ \frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} $ \text{\hspace{1} }\cdot \text{\hspace{1} }\frac{\hspace{2} {\partial}x \hspace{2}}{\hspace{2} {\partial}u \hspace{2}}+\frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\frac{\hspace{2} {{\partial}}^{2}x \hspace{2}}{\hspace{2} {\partial}{u}^{2} \hspace{2}} \} }$ $3$\displaystyle{+ \{ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}u \hspace{2}} $ \frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}y \hspace{2}} $ \text{\hspace{1} }\cdot \text{\hspace{1} }\frac{\hspace{2} {\partial}y \hspace{2}}{\hspace{2} {\partial}u \hspace{2}}+\frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}y \hspace{2}}\frac{\hspace{2} {{\partial}}^{2}y \hspace{2}}{\hspace{2} {\partial}{u}^{2} \hspace{2}} \} }$

条件から， $3$\displaystyle{\text{\hspace{1}}\text{\hspace{1}}\frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\text{\hspace{1}},\text{\hspace{1}}\text{\hspace{1}}\frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}y \hspace{2}}\text{\hspace{1}}\text{\hspace{1}}}$ は共に合成関数であるから，これらを $3$\displaystyle{\text{\hspace{1}}\text{\hspace{1}}u\text{\hspace{1}}\text{\hspace{1}}}$ で偏微分すると，

$3$\displaystyle{ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}u \hspace{2}} $ \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}} $ \frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} $ \cdot \frac{\hspace{2} {\partial}x \hspace{2}}{\hspace{2} {\partial}u \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}} $ \cdot \frac{\hspace{2} {\partial}y \hspace{2}}{\hspace{2} {\partial}u \hspace{2}} }$

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

$3$\displaystyle{ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}u \hspace{2}} $ \frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\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}} $ \cdot \frac{\hspace{2} {\partial}x \hspace{2}}{\hspace{2} {\partial}u \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}} $ \cdot \frac{\hspace{2} {\partial}y \hspace{2}}{\hspace{2} {\partial}u \hspace{2}} }$

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

これらを代入して，

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

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

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

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

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

$3$\displaystyle{=\hspace{1}{f}_{ xx }{ $ \frac{\hspace{2} {\partial}x \hspace{2}}{\hspace{2} {\partial}u \hspace{2}} $ }^{2}+2\hspace{1}{f}_{ yx }\frac{\hspace{2} {\partial}x \hspace{2}}{\hspace{2} {\partial}u \hspace{2}}\frac{\hspace{2} {\partial}y \hspace{2}}{\hspace{2} {\partial}u \hspace{2}}+\hspace{1}{f}_{ yy }{ $ \frac{\hspace{2} {\partial}y \hspace{2}}{\hspace{2} {\partial}u \hspace{2}} $ }^{2}}$ $3$\displaystyle{+\hspace{1}{f}_{x}\frac{\hspace{2} {{\partial}}^{2}x \hspace{2}}{\hspace{2} {\partial}{u}^{2} \hspace{2}}+\hspace{1}{f}_{y}\frac{\hspace{2} {{\partial}}^{2}y \hspace{2}}{\hspace{2} {\partial}{u}^{2} \hspace{2}}}$

$3$\displaystyle{ \hspace{1}{f}_{ yx }=\hspace{1}{f}_{ xy } }$ より（ここを参照）

$3$\displaystyle{=\hspace{1}{f}_{ xx }{ $ \frac{\hspace{2} {\partial}x \hspace{2}}{\hspace{2} {\partial}u \hspace{2}} $ }^{2}+2\hspace{1}{f}_{ xy }\frac{\hspace{2} {\partial}x \hspace{2}}{\hspace{2} {\partial}u \hspace{2}}\frac{\hspace{2} {\partial}y \hspace{2}}{\hspace{2} {\partial}u \hspace{2}}+\hspace{1}{f}_{ yy }{ $ \frac{\hspace{2} {\partial}y \hspace{2}}{\hspace{2} {\partial}u \hspace{2}} $ }^{2}}$ $3$\displaystyle{+\hspace{1}{f}_{x}\frac{\hspace{2} {{\partial}}^{2}x \hspace{2}}{\hspace{2} {\partial}{u}^{2} \hspace{2}}+\hspace{1}{f}_{y}\frac{\hspace{2} {{\partial}}^{2}y \hspace{2}}{\hspace{2} {\partial}{u}^{2} \hspace{2}}}$

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最終更新日： 2023年1月21日