偏微分の基本公式(I)

$3$\displaystyle{f $ x,y $ }$ は $3$\displaystyle{x,y\text{\hspace{1} }}$ を変数とする関数（2変数関数）， $3$\displaystyle{c\text{\hspace{1} }}$ は定数とする．

$3$\displaystyle{ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}cf $ x,y $ =c\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}f $ x,y $ }$ ⇒導出

$3$\displaystyle{\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}} \{ f $ x,y $ \pm g $ x,y $ \} }$ $3$\displaystyle{=\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}f $ x,y $ \pm \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}g $ x,y $ }$ ⇒導出

$3$\displaystyle{\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}} \{ f $ x,y $ \cdot g $ x,y $ \} }$ $3$\displaystyle{= $ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}f \( x,y $ \) \cdot g $ x,y $ +f $ x,y $ $ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}g \( x,y $ \) }$ ⇒導出

$3$\displaystyle{\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}} \{ \frac{\hspace{2} f $ x,y $ \hspace{2}}{\hspace{2} g $ x,y $ \hspace{2}} \} }$ $3$\displaystyle{=\frac{\hspace{2} $ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}f \( x,y $ \) \cdot g $ x,y $ - f $ x,y $ $ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}g \( x,y $ \) \hspace{2}}{\hspace{2} { \{ g $ x,y $ \} }^{2} \hspace{2}}}$ ⇒導出

$3$\displaystyle{ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}f\left({ g\left({ x,y }\right) }\right)={f}^{\prime }\left({ g\left({ x,y }\right) }\right)\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}g\left({ x,y }\right) }$ ⇒導出

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最終更新日： 2025年4月25日