積の微分

\begin{eqnarray*}
(f(x)g(x))' &=& \lim_{h \to 0} \frac{f(x+h)g(x+h) - f(x)g(x)}{h}\\
&=& \lim_{h \to 0} \frac{f(x+h)g(x) - f(x)g(x) + f(x+h)g(x+h) - g(x)f(x + h)}{h}\\
&=& \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}g(x) + \lim_{h\to0} f(x)\frac{g(x+h)-g(x)}{h}\\
&=& f'(x)g(x) + f(x)g'(x)
\end{eqnarray*}