The sequence of functions $f _ { 1 } ( x ) , f _ { 2 } ( x ) , f _ { 3 } ( x ) , \ldots$ is defined as follows:
$$\begin{aligned}
f _ { 1 } ( x ) & = x ^ { 10 } \\
f _ { n + 1 } ( x ) & = x f _ { n } ^ { \prime } ( x ) \text { for } n \geq 1
\end{aligned}$$
where $f _ { n } ^ { \prime } ( x ) = \frac { d f _ { n } ( x ) } { d x }$
Find the value of
$$\sum _ { n = 1 } ^ { 20 } f _ { n } ( x )$$