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.
-
$y = x$ $y = 2 x$

A) Since $\lim _ { x - \theta ^ { + } } f ^ { \prime } ( x ) = 1$ while $\lim _ { x \rightarrow \theta ^ { - } } f ^ { \prime } ( x ) = 2 , f ( x )$ is not differentiable at $x = 0$ .
B) Since $\lim _ { x \rightarrow \theta ^ { + } } f ^ { \prime } ( x ) = 2$ while $\lim _ { x \rightarrow \theta ^ { - } } f ^ { \prime } ( x ) = 1 , f ( x )$ is not differentiable at $x = 0$ .
C) Since $\lim _ { x - \theta ^ { + } } f ^ { \prime } ( x ) = - 2$ while $\lim _ { x - \theta ^ { - } } f ^ { \prime } ( x ) = - 1 , f ( x )$ is not differentiable at $x = 0$ .
D) Since $\lim _ { x \rightarrow \theta ^ { + } } f ^ { \prime } ( x ) = 1$ while $\lim _ { x - \theta ^ { - } } f ^ { \prime } ( x ) = 1 , f ( x )$ is differentiable at $x = 0$ .