For the following functions, find $f(x)$, $c$, $f^\prime(x)$, and $f^\prime(c)$.

1. $f ^ { \prime } ( c ) = \lim\limits_ { \Delta x \rightarrow 0 } \frac { ( 5 - 3 ( 1 + \Delta x ) ) - 2 } { \Delta x }$

   Hint (a):

   This is Hana's Definition of the Derivative. Notice that $Δx$ is equivalent to $h$.

   Answer (a):

   $f(x) = 5 − 3x$ where $c = 1$
   so $f^\prime(x) = −3$ and $f^\prime(1) = −3$

2. $f ^ { \prime } ( c ) = \lim\limits_ { \Delta x \rightarrow 0 } \frac { ( - 2 + \Delta x ) ^ { 3 } + 8 } { \Delta x }$

   Hint (b):

   See hint in part (a).

3. $f ^ { \prime } ( c ) = \lim\limits_ { x \rightarrow 9 } \frac { 2 x ^ { 1 / 2 } - 6 } { x - 9 }$

   Hint (c):

   This is Ana's Definition of the Derivative.