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

Answer

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

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

### Step 2: Apply the power rule
Next, we apply the power rule of differentiation. The power rule states that if \( f(x) = x^n \), then the derivative \( f'(x) \) is given by:
\[ f'(x) = n \cdot x^{n-1} \]
In our case, \( n = -1 \). Therefore, we differentiate:
\[ f'(x) = -1 \cdot x^{-1 - 1} = -1 \cdot x^{-2} \]

### Step 3: Simplify the derivative
Now we simplify the expression for the derivative:
\[ f'(x) = -1 \cdot x^{-2} = -\frac{1}{x^2} \]

### Conclusion
Thus, the derivative of the function \( f(x) = \frac{1}{x} \) is:
\[ f'(x) = -\frac{1}{x^2} \]