偏微分の基本公式(I)の導出：商

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

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

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

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

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

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

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

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

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

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

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