Find, from first principles, the derivative of the following w.r.t. x :
`1/x, x ne 0`

Answer

To find the derivative of the function \( f(x) = \frac{1}{x} \) using first principles, we will follow these steps:

### Step 1: Write the definition of the derivative
The derivative of a function \( f(x) \) at a point \( x \) is given by:
\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]

### Step 2: Substitute the function into the definition
For our function \( f(x) = \frac{1}{x} \), we need to find \( f(x+h) \):
\[ f(x+h) = \frac{1}{x+h} \]
Now substitute this into the derivative formula:
\[ f'(x) = \lim_{h \to 0} \frac{\frac{1}{x+h} - \frac{1}{x}}{h} \]

### Step 3: Simplify the expression
To simplify the expression in the limit, we need a common denominator for the fractions in the numerator:
\[ f'(x) = \lim_{h \to 0} \frac{\frac{x - (x+h)}{(x+h)x}}{h} \]
This simplifies to:
\[ f'(x) = \lim_{h \to 0} \frac{\frac{-h}{(x+h)x}}{h} \]

### Step 4: Cancel \( h \)
We can cancel \( h \) in the numerator and denominator:
\[ f'(x) = \lim_{h \to 0} \frac{-1}{(x+h)x} \]

### Step 5: Apply the limit
Now, we can apply the limit as \( h \) approaches 0:
\[ f'(x) = \frac{-1}{(x+0)x} = \frac{-1}{x^2} \]

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

---