### Home > APCALC > Chapter 5 > Lesson 5.2.4 > Problem5-99

5-99.

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

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

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

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

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

See hint in part (a).

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

This is Ana's Definition of the Derivative.