# 分数関数の微分 II （関数の商の導関数）

${\left\{\frac{h\left(x\right)}{g\left(x\right)}\right\}}^{\prime }=\frac{{h}^{\prime }\left(x\right)g\left(x\right)-h\left(x\right){g}^{\prime }\left(x\right)}{{\left\{g\left(x\right)\right\}}^{2}}$

すなわち

$f\left(x\right)=\frac{h\left(x\right)}{g\left(x\right)}$ $\to {f}^{\prime }\left(x\right)=\frac{{h}^{\prime }\left(x\right)g\left(x\right)-h\left(x\right){g}^{\prime }\left(x\right)}{{\left\{g\left(x\right)\right\}}^{2}}$

## ■導出

${f}^{\prime }\left(x\right)=\underset{\Delta x\to 0}{lim}\frac{f\left(x+\Delta x\right)-f\left(x\right)}{\Delta x}$

$=\underset{\Delta x\to 0}{lim}\frac{\frac{h\left(x+\Delta x\right)}{g\left(x+\Delta x\right)}-\frac{h\left(x\right)}{g\left(x\right)}}{\Delta x}$

$=\underset{\Delta x\to 0}{lim}\frac{\frac{h\left(x+\Delta x\right)g\left(x\right)-h\left(x\right)g\left(x+\Delta x\right)}{g\left(x+\Delta x\right)g\left(x\right)}}{\Delta x}$

• $=\underset{\Delta x\to 0}{lim}\left\{\frac{1}{g\left(x+h\right)g\left(x\right)}·$

• $\frac{h\left(x+\Delta x\right)g\left(x\right)-h\left(x\right)g\left(x+\Delta x\right)}{\Delta x}\right\}$

$=\left\{\underset{\Delta x\to 0}{lim}\frac{1}{g\left(x+\Delta x\right)g\left(x\right)}\right\}$ $×\left\{\underset{\Delta x\to 0}{\mathrm{lim}}\frac{h\left(x+\Delta x\right)g\left(x\right)-h\left(x\right)g\left(x\right)+h\left(x\right)g\left(x\right)-h\left(x\right)g\left(x+\Delta x\right)}{\Delta x}\right\}$

$=\left\{\underset{\Delta x\to 0}{lim}\frac{1}{g\left(x+\Delta x\right)g\left(x\right)}\right\}$$×\left[\underset{\Delta x\to 0}{\mathrm{lim}}\frac{\left\{h\left(x+\Delta x\right)-h\left(x\right)\right\}g\left(x\right)+h\left(x\right)\left\{g\left(x\right)-g\left(x+\Delta x\right)\right\}}{\Delta x}\right]$

$=\left\{\underset{\Delta x\to 0}{lim}\frac{1}{g\left(x+\Delta x\right)g\left(x\right)}\right\}$$×\left[\left\{\underset{\Delta x\to 0}{\mathrm{lim}}\frac{h\left(x+\Delta x\right)-h\left(x\right)}{\Delta x}\right\}g\left(x\right)-h\left(x\right)\left\{\underset{\Delta x\to 0}{\mathrm{lim}}\frac{g\left(x+\Delta x\right)-g\left(x\right)}{\Delta x}\right\}\right]$

$=\frac{{h}^{\prime }\left(x\right)g\left(x\right)-h\left(x\right){g}^{\prime }\left(x\right)}{{\left\{g\left(x\right)\right\}}^{2}}$

ここを参照

よって

${\left\{\frac{h\left(x\right)}{g\left(x\right)}\right\}}^{\prime }=\frac{{h}^{\prime }\left(x\right)g\left(x\right)-h\left(x\right){g}^{\prime }\left(x\right)}{{\left\{g\left(x\right)\right\}}^{2}}$

である．

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