$3$\displaystyle{ \frac{\hspace{2}d\hspace{2}}{\hspace{2} dx \hspace{2}}\int _{a}^{x} f $t$ dt =f $x$ }$

$3$\displaystyle{ F$t$ }$ を $3$\displaystyle{ f$t$ }$ の原始関数とすると
$3$\displaystyle{ \displaystyle{ \int _{a}^{x} f$t$dt } }$
　 $3$\displaystyle{ =F$x$- F$a$ }$
$3$\displaystyle{\frac{\hspace{2}d\hspace{2}}{\hspace{2} dx \hspace{2}}\displaystyle{ \int _{a}^{x} f$t$dt }}$
　 $3$\displaystyle{=\frac{\hspace{2}d\hspace{2}}{\hspace{2} dx \hspace{2}} \{ F$x$- F$a$ \} }$
　 $3$\displaystyle{=\frac{\hspace{2}d\hspace{2}}{\hspace{2} dx \hspace{2}}F$x$- \frac{\hspace{2}d\hspace{2}}{\hspace{2} dx \hspace{2}}F$a$}$
$3$\displaystyle{F$a$}$ は定数なので
$3$\displaystyle{\frac{\hspace{2}d\hspace{2}}{\hspace{2} dx \hspace{2}}F$a$}$ は $3$\displaystyle{0}$ .したがって,
　 $3$\displaystyle{=f$x$}$ 　　　　　　　　　