Suppose $f : \mathbb { R } \rightarrow \mathbb { R }$ is a function given by
$$f ( x ) = \begin{cases} 1 & \text { if } x = 1 \\ e ^ { \left( x ^ { 10 } - 1 \right) } + ( x - 1 ) ^ { 2 } \sin \left( \frac { 1 } { x - 1 } \right) & \text { if } x \neq 1 \end{cases}$$
(a) Find $f ^ { \prime } ( 1 )$.
(b) Evaluate $\lim _ { u \rightarrow \infty } \left[ 100 u - u \sum _ { k = 1 } ^ { 100 } f \left( 1 + \frac { k } { u } \right) \right]$.