商の微分
$3$\displaystyle{ { \{ \frac{\hspace{2} f $x$ \hspace{2}}{\hspace{2} g $x$ \hspace{2}} \} }^{\prime }=\frac{\hspace{2} {f}^{\prime } $x$ g $x$ - f $x$ {g}^{\prime } $x$ \hspace{2}}{\hspace{2} { \{ g $x$ \} }^{2} \hspace{2}} }$

$3$\displaystyle{\begin{array}{lll}{f}^{\prime } $x$ ={ \lim }\limits_{ h\rightarrow 0 }\frac{\hspace{2} f $ x+h $ - f $x$ \hspace{2}}{\hspace{2}h\hspace{2}}& \vspace{6}\\ ={ \lim }\limits_{ h\rightarrow 0 }\frac{\hspace{2} \frac{\hspace{2} h $ x+h $ \hspace{2}}{\hspace{2} g $ x+h $ \hspace{2}}- \frac{\hspace{2} h $x$ \hspace{2}}{\hspace{2} g $x$ \hspace{2}} \hspace{2}}{\hspace{2}h\hspace{2}}& \vspace{6}\\ ={ \lim }\limits_{ h\rightarrow 0 }\frac{\hspace{2} \frac{\hspace{2} h $ x+h $ g $x$ - h $x$ g $ x+h $ \hspace{2}}{\hspace{2} g $ x+h $ g $x$ \hspace{2}} \hspace{2}}{\hspace{2}h\hspace{2}}& \vspace{6}\\ \end{array}}$
$3$\displaystyle{\begin{array}{lll}={ \lim }\limits_{ h\rightarrow 0 } \{\begin{array}{lll}\frac{\hspace{2}1\hspace{2}}{\hspace{2} g $ x+h $ g $x$ \hspace{2}}\cdot & \vspace{6}\\ \frac{\hspace{2} h $ x+h $ g $x$ - h $x$ g $ x+h $ \hspace{2}}{\hspace{2} g $ x+h $ g $x$ \hspace{2}}& \vspace{6}\\ \end{array}\} & \vspace{6}\\ = \{ { \lim }\limits_{ h\rightarrow 0 }\frac{\hspace{2}1\hspace{2}}{\hspace{2} g $ x+h $ g $x$ \hspace{2}} \} & \vspace{6}\\ \{ { \lim }\limits_{ h\rightarrow 0 } \[\begin{array}{lll}\frac{\hspace{2} h $ x+h $ g $x$ - h $x$ g $x$ \hspace{2}}{\hspace{2}h\hspace{2}}& \vspace{6}\\ \frac{\hspace{2} h $x$ g $x$ - h $x$ g $ x+h $ \hspace{2}}{\hspace{2}h\hspace{2}}& \vspace{6}\\ \end{array}\] \} & \vspace{6}\\ \end{array}}$

$3$\displaystyle{\begin{array}{lll}= \{ { \lim }\limits_{ h\rightarrow 0 }\frac{\hspace{2}1\hspace{2}}{\hspace{2} g $ x+h $ g $x$ \hspace{2}} \} & \vspace{6}\\ \{ { \lim }\limits_{ h\rightarrow 0 } \[\begin{array}{lll}\frac{\hspace{2} \[ h $ x+h $ - h $x$ \] g $x$ \hspace{2}}{\hspace{2}h\hspace{2}}& \vspace{6}\\ +\frac{\hspace{2} h $x$ \[ g $x$ - g $ x+h $ \] \hspace{2}}{\hspace{2}h\hspace{2}}& \vspace{6}\\ \end{array}\] \} & \vspace{6}\\ \end{array}}$

$3$\displaystyle{\begin{array}{lll}= \{ { \lim }\limits_{ h\rightarrow 0 }\frac{\hspace{2}1\hspace{2}}{\hspace{2} g $ x+h $ g $x$ \hspace{2}} \} & \vspace{6}\\ \{\begin{array}{lll} \[ { \lim }\limits_{ h\rightarrow 0 }\frac{\hspace{2} h $ x+h $ - h $x$ \hspace{2}}{\hspace{2}h\hspace{2}} \] g $x$ & \vspace{6}\\ - h $x$ \[ { \lim }\limits_{ h\rightarrow 0 }\frac{\hspace{2} g $ x+h $ - g $x$ \hspace{2}}{\hspace{2}h\hspace{2}} \] & \vspace{6}\\ \end{array}\} & \vspace{6}\\ \end{array}}$

$3$\displaystyle{=\frac{\hspace{2} {h}^{\prime } $x$ g $x$ - h $x$ {g}^{\prime } $x$ \hspace{2}}{\hspace{2} g{ $x$ }^{2} \hspace{2}}}$
　　　　　　（ここを参照⇒）

よって,
$3$\displaystyle{{ \{ \frac{\hspace{2} h $x$ \hspace{2}}{\hspace{2} g $x$ \hspace{2}} \} }^{\prime }=\frac{\hspace{2} {h}^{\prime } $x$ g $x$ - h $x$ {g}^{\prime } $x$ \hspace{2}}{\hspace{2} g{ $x$ }^{2} \hspace{2}}}$
である。

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