合成関数の偏導関数の導出

2変数関数 $3$\displaystyle{z=f $ x,y $ }$ で $3$\displaystyle{x={\varphi} $t$ ,y={\psi} $t$ }$ ならば，偏導関数 $3$\displaystyle{\frac{\hspace{2} dz \hspace{2}}{\hspace{2} dt \hspace{2}}}$ は

$3$\displaystyle{\frac{\hspace{2} dz \hspace{2}}{\hspace{2} dt \hspace{2}}=\hspace{1}{f}_{x}\frac{\hspace{2} dx \hspace{2}}{\hspace{2} dt \hspace{2}}+\hspace{1}{f}_{y}\frac{\hspace{2} dy \hspace{2}}{\hspace{2} dt \hspace{2}}}$

となる．

■導出

$3$\displaystyle{ \frac{\hspace{2} dz \hspace{2}}{\hspace{2} dt \hspace{2}} }$ $3$\displaystyle{ ={ \lim }\limits_{ h\rightarrow 0 }\frac{\hspace{2} f $ {\varphi} \( t+h $ ,{\psi} $ t+h $ \) - f $ {\varphi} \(t$ ,{\psi} $t$ \) \hspace{2}}{\hspace{2}h\hspace{2}} }$

$3$\displaystyle{={ \lim }\limits_{ h\rightarrow 0 }\frac{\hspace{2} f $ {\varphi} \( t+h $ ,{\psi} $ t+h $ \) - f $ {\varphi} \(t$ ,{\psi} $ t+h $ \) +f $ {\varphi} \(t$ ,{\psi} $ t+h $ \) - f $ {\varphi} \(t$ ,{\psi} $t$ \) \hspace{2}}{\hspace{2}h\hspace{2}}}$

$3$\displaystyle{={ \lim }\limits_{ h\rightarrow 0 }\frac{\hspace{2} f $ {\varphi} \( t+h $ ,{\psi} $ t+h $ \) - f $ {\varphi} \(t$ ,{\psi} $ t+h $ \) \hspace{2}}{\hspace{2}h\hspace{2}}}$

$3$\displaystyle{={ \lim }\limits_{ h\rightarrow 0 }\frac{\hspace{2} f $ {\varphi} \( t+h $ ,{\psi} $ t+h $ \) - f $ {\varphi} \(t$ ,{\psi} $ t+h $ \) \hspace{2}}{\hspace{2} {\varphi} $ t+h $ - {\varphi} $t$ \hspace{2}}\frac{\hspace{2} {\varphi} $ t+h $ - {\varphi} $t$ \hspace{2}}{\hspace{2}h\hspace{2}}}$ $3$\displaystyle{ +{ \lim }\limits_{ h\rightarrow 0 }\frac{\hspace{2} f $ {\varphi} \(t$ ,{\psi} $ t+h $ \) - f $ {\varphi} \(t$ ,{\psi} $t$ \) \hspace{2}}{\hspace{2} {\psi} $ t+h $ - {\psi} $t$ \hspace{2}}\frac{\hspace{2} {\psi} $ t+h $ - {\psi} $t$ \hspace{2}}{\hspace{2}h\hspace{2}} }$

$3$\displaystyle{=\frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\frac{\hspace{2} dx \hspace{2}}{\hspace{2} dt \hspace{2}}+\frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}y \hspace{2}}\frac{\hspace{2} dy \hspace{2}}{\hspace{2} dt \hspace{2}}}$

$3$\displaystyle{=\hspace{1}{f}_{x}\frac{\hspace{2} dx \hspace{2}}{\hspace{2} dt \hspace{2}}+\hspace{1}{f}_{y}\frac{\hspace{2} dy \hspace{2}}{\hspace{2} dt \hspace{2}}}$

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最終更新日： 2024年5月15日