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# 分数関数の微分 II （関数の商の導関数）

$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}}}$

すなわち，

$3\displaystyle{ f \(x\) =\frac{\hspace{2} h \(x\) \hspace{2}}{\hspace{2} g \(x\) \hspace{2}} }$ $3\displaystyle{ \rightarrow {f}^{\prime } \(x\) =\frac{\hspace{2} {h}^{\prime } \(x\) g \(x\) - h \(x\) {g}^{\prime } \(x\) \hspace{2}}{\hspace{2} { \{ g \(x\) \} }^{2} \hspace{2}} }$

## ■導出

$3\displaystyle{ {f}^{\prime } \(x\) ={\lim}\limits_{ h\rightarrow 0 }\frac{\hspace{2} f \( x+ h \) - f \(x\) \hspace{2}}{\hspace{2}h\hspace{2}} }$

$3\displaystyle{ ={\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}} }$

$3\displaystyle{ ={\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}} }$

• $3\displaystyle{\displaystyle{={\lim}\limits_{ h\rightarrow 0 } \{ \frac{\hspace{2}1\hspace{2}}{\hspace{2} g \( x+ h \) g \(x\) \hspace{2}}\cdot }}$

• $3\displaystyle{\displaystyle{ \frac{\hspace{2} h \( x+ h \) g \(x\) - h \(x\) g \( x+ h \) \hspace{2}}{\hspace{2}h\hspace{2}} \} }}$

$3\displaystyle{ = \{ {\lim}\limits_{ h\rightarrow 0 }\frac{\hspace{2}1\hspace{2}}{\hspace{2} g \( x+ h \) g \(x\) \hspace{2}} \} }$$3\displaystyle{ \left{{ {\lim}\limits_{ h\rightarrow 0 }\frac{\hspace{2}1\hspace{2}}{\hspace{2}h\hspace{2}}\left({ h \( x+ h \) g \(x\) - h \(x\) g \(x\) }$$3\displaystyle{ \left.{ \left.{ + h \(x\) g \(x\) - h \(x\) g \( x+ h \) }\right\)\begin{array}\hspace{2}& \vspace{6}\\ \hspace{2}& \vspace{6}\\ \end{array} }\right\} }$

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

$3\displaystyle{ = \{ {\lim}\limits_{ h\rightarrow 0 }\frac{\hspace{2}1\hspace{2}}{\hspace{2} g \( x+ h \) g \(x\) \hspace{2}} \} }$$3\displaystyle{ \left[{ \{ {\lim}\limits_{ h\rightarrow 0 }\frac{\hspace{2} h \( x+ h \) - h \(x\) \hspace{2}}{\hspace{2}h\hspace{2}} \} g \(x\) }$$3\displaystyle{ \left.{ - h \(x\) \{ {\lim}\limits_{ h\rightarrow 0 }\frac{\hspace{2} g \( x+ h \) - g \(x\) \hspace{2}}{\hspace{2}h\hspace{2}} \} }\right\] }$

$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．（分数関数の微分Iを参照の）

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