If f(x) =|x| , then f'(0) is

Options

A. 0
B. 1
C. `-1`
D. Not in existence

Answer

To find \( f'(0) \) for the function \( f(x) = |x| \), we need to determine the left-hand and right-hand derivatives at \( x = 0 \).

### Step 1: Define the function
The function is given as:
\[ f(x) = |x| \]

### Step 2: Calculate the left-hand derivative at \( x = 0 \)
The left-hand derivative is defined as:
\[ f'_{-}(0) = \lim_{h \to 0^-} \frac{f(0 + h) - f(0)}{h} \]
Substituting \( f(0) = |0| = 0 \):
\[ f'_{-}(0) = \lim_{h \to 0^-} \frac{|h| - 0}{h} = \lim_{h \to 0^-} \frac{-h}{h} = \lim_{h \to 0^-} -1 = -1 \]

### Step 3: Calculate the right-hand derivative at \( x = 0 \)
The right-hand derivative is defined as:
\[ f'_{+}(0) = \lim_{h \to 0^+} \frac{f(0 + h) - f(0)}{h} \]
Again substituting \( f(0) = 0 \):
\[ f'_{+}(0) = \lim_{h \to 0^+} \frac{|h| - 0}{h} = \lim_{h \to 0^+} \frac{h}{h} = \lim_{h \to 0^+} 1 = 1 \]

### Step 4: Compare the left-hand and right-hand derivatives
We have:
\[ f'_{-}(0) = -1 \quad \text{and} \quad f'_{+}(0) = 1 \]
Since the left-hand derivative does not equal the right-hand derivative:
\[ f'_{-}(0) \neq f'_{+}(0) \]

### Conclusion
Thus, the derivative \( f'(0) \) does not exist.

\[ \text{Therefore, } f'(0) \text{ does not exist.} \]

### Final Answer
The answer is: \( f'(0) \) does not exist.

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