Let $f ( x )$ be a real differentiable function such that $f ( 0 ) = 1$ and $f ( x + y ) = f ( x ) f ^ { \prime } ( y ) + f ^ { \prime } ( x ) f ( y )$ for all $x , y \in \mathbf { R }$. Then $\sum _ { \mathrm { n } = 1 } ^ { 100 } \log _ { \mathrm { e } } f ( \mathrm { n } )$ is equal to:\\
(1) 2525\\
(2) 5220\\
(3) 2384\\
(4) 2406