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# 合成関数の偏導関数の導出

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

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

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}}+{ \lim }\limits_{ h\rightarrow 0 }\frac{\hspace{2} 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} {\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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