The derivative of `f(x) =|x|` at x = 0 is:

Options

A. 1
B. 0
C. `-1`
D. does not exist

Answer

To find the derivative of the function \( f(x) = |x| \) at \( x = 0 \), we will follow these steps:

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

### Step 2: Find the left-hand derivative at \( x = 0 \)
To find the left-hand derivative, we consider \( x \) approaching \( 0 \) from the left (i.e., \( x < 0 \)):
\[ f(x) = -x \]
The derivative of \( f(x) \) when \( x < 0 \) is:
\[ f'(x) = \frac{d}{dx}(-x) = -1 \]
Thus, the left-hand derivative at \( x = 0 \) is:
\[ f'_{-}(0) = -1 \]

### Step 3: Find the right-hand derivative at \( x = 0 \)
Now, we consider \( x \) approaching \( 0 \) from the right (i.e., \( x \geq 0 \)):
\[ f(x) = x \]
The derivative of \( f(x) \) when \( x \geq 0 \) is:
\[ f'(x) = \frac{d}{dx}(x) = 1 \]
Thus, the right-hand derivative at \( x = 0 \) is:
\[ f'_{+}(0) = 1 \]

### Step 4: Compare the left-hand and right-hand derivatives
We have:
- Left-hand derivative: \( f'_{-}(0) = -1 \)
- Right-hand derivative: \( f'_{+}(0) = 1 \)

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

### Final Answer
The derivative of \( f(x) = |x| \) at \( x = 0 \) is:
\[ \text{Does not exist} \]

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