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 for different intervals
The absolute value function \( |x| \) behaves differently based on whether \( x \) is positive or negative. Therefore, we can express \( f(x) \) as follows:

- For \( x < 0 \):
\[ f(x) = |x|^3 = (-x)^3 = -x^3 \]

- For \( x \geq 0 \):
\[ f(x) = |x|^3 = x^3 \]

### Step 2: Differentiate the function for each interval
Next, we differentiate \( f(x) \) for both cases.

- For \( x < 0 \):
\[ f'(x) = \frac{d}{dx}(-x^3) = -3x^2 \]

- For \( x \geq 0 \):
\[ f'(x) = \frac{d}{dx}(x^3) = 3x^2 \]

### Step 3: Calculate the left-hand derivative (LHD) at \( x = 0 \)
To find the derivative at \( x = 0 \), we first compute the left-hand derivative (LHD):

\[ \text{LHD} = \lim_{x \to 0^-} f'(x) = \lim_{x \to 0^-} (-3x^2) \]

Substituting \( x = 0 \):

\[ \text{LHD} = -3(0)^2 = 0 \]

### Step 4: Calculate the right-hand derivative (RHD) at \( x = 0 \)
Next, we compute the right-hand derivative (RHD):

\[ \text{RHD} = \lim_{x \to 0^+} f'(x) = \lim_{x \to 0^+} (3x^2) \]

Substituting \( x = 0 \):

\[ \text{RHD} = 3(0)^2 = 0 \]

### Step 5: Compare LHD and RHD
Since both the left-hand derivative and the right-hand derivative at \( x = 0 \) are equal:

\[ \text{LHD} = \text{RHD} = 0 \]

### Conclusion
Thus, the derivative of \( f(x) = |x|^3 \) at \( x = 0 \) exists and is equal to:

\[ f'(0) = 0 \]

### Final Answer
The derivative of \( f(x) = |x|^3 \) at \( x = 0 \) is \( 0 \).

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