積の微分

積の微分

$$\begin{eqnarray} \frac{\mathrm{d}\{f(x)g(x)\}}{\mathrm{d}t}&=&\displaystyle \lim_{h \to 0}\frac{f(x+h)g(x+h)-f(x)g(x)}{h}\nonumber\\ &=&\lim_{h \to 0}\frac{f(x+h)g(x+h)-f(x+h)g(x)+f(x+h)g(x)-f(x)g(x)}{h}\;\dots\;-f(x+h)g(x)+f(x+h)g(x)=0\nonumber\\ &=& \lim_{h \to 0}\left\{\frac{g(x+h)-g(x)}{h}f(x+h) + \frac{f(x+h)-f(x)}{h}g(x)\right\} \nonumber\\ &=&\lim_{h \to 0}\frac{g(x+h)-g(x)}{h}f(x+h) + \lim_{h \to 0}\frac{f(x+h)-f(x)}{h}g(x) \nonumber\\ &=&\frac{\mathrm{d}\{g(x)\}}{\mathrm{d}t}f(x) + \frac{\mathrm{d}\{f(x)\}}{\mathrm{d}t}g(x) \nonumber\\ \end{eqnarray}$$ $$ (fg)'=f'g+fg'\\ $$