The derivative of `f(x)=|x|^(3) at x=0,` is

Options

A. `-1`
B. 0
C. does not exist
D. None of these

Answer

To find the derivative of the function \( f(x) = |x|^3 \) at \( x = 0 \), we will follow these steps:

### Step 1: Define the function
The function is given as:
\[ f(x) = |x|^3 \]

### Step 2: Determine the piecewise definition
The absolute value function can be expressed in a piecewise manner:
\[ f(x) = \begin{cases} x^3 & \text{if } x \geq 0 \\ -x^3 & \text{if } x < 0 \end{cases} \]

### Step 3: Find the derivative for \( x > 0 \) and \( x < 0 \)
Now we differentiate \( f(x) \) in both cases:
- For \( x > 0 \):
\[ f'(x) = 3x^2 \]
- For \( x < 0 \):
\[ f'(x) = -3x^2 \]

### Step 4: Evaluate the derivative at \( x = 0 \)
To find the derivative at \( x = 0 \), we need to check the limits from both sides:
- Right-hand limit as \( x \) approaches 0:
\[ \lim_{x \to 0^+} f'(x) = \lim_{x \to 0^+} 3x^2 = 3(0)^2 = 0 \]
- Left-hand limit as \( x \) approaches 0:
\[ \lim_{x \to 0^-} f'(x) = \lim_{x \to 0^-} -3x^2 = -3(0)^2 = 0 \]

### Step 5: Conclude the derivative at \( x = 0 \)
Since both the right-hand limit and left-hand limit at \( x = 0 \) are equal:
\[ f'(0) = 0 \]

Thus, the derivative of \( f(x) = |x|^3 \) at \( x = 0 \) is:
\[ \boxed{0} \]

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