Find the derivatives of the following at any point of their domains :
`f(x)=1/(sqrt(3)x^(3)), x ne 0`

Answer

To find the derivative of the function \( f(x) = \frac{1}{\sqrt{3} x^3} \) where \( x \neq 0 \), we can follow these steps:

### Step 1: Rewrite the function
We can rewrite the function in a form that is easier to differentiate. The function can be expressed as:
\[ f(x) = \frac{1}{\sqrt{3}} \cdot x^{-3} \]

### Step 2: Differentiate the function
Now, we will differentiate \( f(x) \) using the power rule. The power rule states that if \( f(x) = x^n \), then \( f'(x) = n \cdot x^{n-1} \).

Applying the power rule here:
\[ f'(x) = \frac{1}{\sqrt{3}} \cdot \frac{d}{dx}(x^{-3}) = \frac{1}{\sqrt{3}} \cdot (-3) \cdot x^{-4} \]

### Step 3: Simplify the derivative
Now, we simplify the expression:
\[ f'(x) = -\frac{3}{\sqrt{3}} \cdot x^{-4} \]
We can simplify \(-\frac{3}{\sqrt{3}}\) to \(-\sqrt{3}\):
\[ f'(x) = -\sqrt{3} \cdot x^{-4} \]

### Step 4: Rewrite in standard form
Finally, we can rewrite the derivative in a more standard form:
\[ f'(x) = -\frac{\sqrt{3}}{x^4} \]

### Final Answer
Thus, the derivative of the function \( f(x) = \frac{1}{\sqrt{3} x^3} \) is:
\[ f'(x) = -\frac{\sqrt{3}}{x^4} \]

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