偏微分の基本公式(I)の導出：積

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

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

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

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

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

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

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

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

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