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

Options

A. `-1`
B. not defined
C. 0
D. `1//2`

Answer

To find the derivative of the function \( f(x) = |x|^3 \) at \( x = 0 \), we can use the definition of the derivative. The derivative at a point \( x = a \) is defined as:

\[ f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} \]

In our case, we want to find \( f'(0) \).

### Step 1: Define the function
Since \( f(x) = |x|^3 \), we can express it piecewise:
- For \( x < 0 \), \( f(x) = (-x)^3 = -x^3 \)
- For \( x \geq 0 \), \( f(x) = x^3 \)

### Step 2: Calculate the left-hand derivative at \( x = 0 \)
The left-hand derivative is given by:

\[ f'_{-}(0) = \lim_{h \to 0^-} \frac{f(0 + h) - f(0)}{h} \]

Since \( f(0) = |0|^3 = 0 \), we have:

\[ f'_{-}(0) = \lim_{h \to 0^-} \frac{f(h)}{h} = \lim_{h \to 0^-} \frac{(-h)^3}{h} = \lim_{h \to 0^-} \frac{-h^3}{h} = \lim_{h \to 0^-} -h^2 = 0 \]

### Step 3: Calculate the right-hand derivative at \( x = 0 \)
The right-hand derivative is given by:

\[ f'_{+}(0) = \lim_{h \to 0^+} \frac{f(0 + h) - f(0)}{h} \]

Again, since \( f(0) = 0 \):

\[ f'_{+}(0) = \lim_{h \to 0^+} \frac{f(h)}{h} = \lim_{h \to 0^+} \frac{h^3}{h} = \lim_{h \to 0^+} h^2 = 0 \]

### Step 4: Conclude the derivative at \( x = 0 \)
Since both the left-hand and right-hand derivatives at \( x = 0 \) are equal:

\[ f'_{-}(0) = f'_{+}(0) = 0 \]

Thus, the derivative of \( f(x) = |x|^3 \) at \( x = 0 \) is:

\[ f'(0) = 0 \]

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

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