The derivative of `|x|` at x = 0

Options

A. is 1
B. is `-1`
C. is 0
D. does nto exist

Explanation

To find the derivative of \( |x| \) at \( x = 0 \), we will analyze the function and its behavior around that point.

### Step 1: Define the function
The absolute value function \( |x| \) can be defined piecewise:
\[ |x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases} \]

### Step 2: Find the right-hand derivative at \( x = 0 \)
The right-hand derivative at \( x = 0 \) is calculated using the definition of the derivative:
\[ f'(0) = \lim_{h \to 0^+} \frac{f(0+h) - f(0)}{h} \]
Substituting \( f(x) = |x| \):
\[ f'(0) = \lim_{h \to 0^+} \frac{|h| - |0|}{h} = \lim_{h \to 0^+} \frac{h - 0}{h} = \lim_{h \to 0^+} 1 = 1 \]

### Step 3: Find the left-hand derivative at \( x = 0 \)
The left-hand derivative at \( x = 0 \) is calculated similarly:
\[ f'(0) = \lim_{h \to 0^-} \frac{f(0+h) - f(0)}{h} \]
Substituting \( f(x) = |x| \):
\[ f'(0) = \lim_{h \to 0^-} \frac{|h| - |0|}{h} = \lim_{h \to 0^-} \frac{-h - 0}{h} = \lim_{h \to 0^-} -1 = -1 \]

### Step 4: Compare the right-hand and left-hand derivatives
From the calculations:
- Right-hand derivative at \( x = 0 \) is \( 1 \)
- Left-hand derivative at \( x = 0 \) is \( -1 \)

Since the right-hand derivative and left-hand derivative at \( x = 0 \) are not equal, we conclude that the derivative of \( |x| \) at \( x = 0 \) does not exist.

### Final Answer
The derivative of \( |x| \) at \( x = 0 \) does not exist.

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