# 偏微分の基本公式(I)の導出：商

$f\left(x,y\right)$$g\left(x,y\right)$$x,y$ を変数とする関数（2変数関数）とすると

$\frac{\partial }{\partial x}\left\{\frac{f\left(x,y\right)}{g\left(x,y\right)}\right\}$$=\frac{\left(\frac{\partial }{\partial x}f\left(x,y\right)\right)g\left(x,y\right)-f\left(x,y\right)\left(\frac{\partial }{\partial x}g\left(x,y\right)\right)}{{\left\{g\left(x,y\right)\right\}}^{2}}$

が成り立つ．

## ■導出

$\frac{\partial }{\partial x}f\left(x,y\right)=\underset{h\to 0}{\mathrm{lim}}\frac{f\left(x+h,y\right)-f\left(x,y\right)}{h}$

を利用すると

$\frac{\partial }{\partial x}\left\{\frac{f\left(x,y\right)}{g\left(x,y\right)}\right\}$

$=\underset{h\to 0}{\mathrm{lim}}\frac{\left\{\frac{f\left(x+h,y\right)}{g\left(x+h,y\right)}\right\}-\left\{\frac{f\left(x,y\right)}{g\left(x,y\right)}\right\}}{h}$

$=\underset{h\to 0}{\mathrm{lim}}\frac{\frac{f\left(x+h,y\right)g\left(x,y\right)-f\left(x,y\right)g\left(x+h,y\right)}{g\left(x,y\right)g\left(x+h,y\right)}}{h}$

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

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

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

$=\frac{1}{{\left\{g\left(x,y\right)\right\}}^{2}}\underset{h\to 0}{\mathrm{lim}}\frac{f\left(x+h,y\right)g\left(x,y\right)-f\left(x,y\right)g\left(x,y\right)+f\left(x,y\right)g\left(x,y\right)-f\left(x,y\right)g\left(x+h,y\right)}{h}$

$=\frac{1}{{\left\{g\left(x,y\right)\right\}}^{2}}\left\{\underset{h\to 0}{\mathrm{lim}}\frac{f\left(x+h,y\right)g\left(x,y\right)-f\left(x,y\right)g\left(x,y\right)}{h}$$+\underset{h\to 0}{\mathrm{lim}}\frac{f\left(x,y\right)g\left(x,y\right)-f\left(x,y\right)g\left(x+h,y\right)}{h}\right\}$

$=\frac{1}{{\left\{g\left(x,y\right)\right\}}^{2}}\left\{\underset{h\to 0}{\mathrm{lim}}\frac{f\left(x+h,y\right)-f\left(x,y\right)}{h}g\left(x,y\right)$$+\underset{h\to 0}{\mathrm{lim}}f\left(x,y\right)\frac{g\left(x,y\right)-g\left(x+h,y\right)}{h}\right\}$

$=\frac{1}{{\left\{g\left(x,y\right)\right\}}^{2}}\left\{g\left(x,y\right)\underset{h\to 0}{\mathrm{lim}}\frac{f\left(x+h,y\right)-f\left(x,y\right)}{h}$$-f\left(x,y\right)\underset{h\to 0}{\mathrm{lim}}\frac{g\left(x+h,y\right)-g\left(x,y\right)}{h}\right\}$

$=\frac{1}{{\left\{g\left(x,y\right)\right\}}^{2}}\left\{\left(\frac{\partial }{\partial x}f\left(x,y\right)\right)g\left(x,y\right)$$-f\left(x,y\right)\left(\frac{\partial }{\partial x}g\left(x,y\right)\right)\right\}$

$=\frac{\left(\frac{\partial }{\partial x}f\left(x,y\right)\right)g\left(x,y\right)-f\left(x,y\right)\left(\frac{\partial }{\partial x}g\left(x,y\right)\right)}{{\left\{g\left(x,y\right)\right\}}^{2}}$

よって

$\frac{\partial }{\partial x}\left\{\frac{f\left(x,y\right)}{g\left(x,y\right)}\right\}$$=\frac{\left(\frac{\partial }{\partial x}f\left(x,y\right)\right)g\left(x,y\right)-f\left(x,y\right)\left(\frac{\partial }{\partial x}g\left(x,y\right)\right)}{{\left\{g\left(x,y\right)\right\}}^{2}}$

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