Compare the Right-Hand and Left-Hand Derivatives to Determine Whether or Not

Multiple Choice

Compare the right-hand and left-hand derivatives to determine whether or not the function is differentiable at the point whose coordinates are given.
- $$y = \frac { 1 } { x } \quad y = - 1$$

A) Since $\lim _ { x \rightarrow 1 ^ { + } } f ^ { \prime } ( x ) = 0$ while $\lim _ { x \rightarrow 1 ^ { - } } f ^ { \prime } ( x ) = 1$ , $f ( x )$ is not differentiable at $x = - 1$ .
B) Since $\lim _ { x \rightarrow 1 ^ { + } } f ^ { \prime } ( x ) = - 1$ while $\lim _ { x \rightarrow 1 ^ { - } } f ^ { \prime } ( x ) = 0 , f ( x )$ is not differentiable at $x = - 1$ .
C) Since $\lim _ { x \rightarrow 1 ^ { + } } f ^ { \prime } ( x ) = 0$ while $\lim _ { x \rightarrow 1 ^ { - } } f ^ { \prime } ( x ) = - 1 , f ( x )$ is not differentiable at $x = - 1$ .
D) Since $\lim _ { x \rightarrow 1 ^ { + } } f ^ { \prime } ( x ) = 0$ while $\lim _ { x \rightarrow 1 ^ { - } } f ^ { \prime } ( x ) = 0$ , $f ( x )$ is differentiable at $x = - 1$ .