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

Question

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

Quizplus · Accepted Answer

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

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