Differentiate the following from ab-initio (or from definition) :
`(x^(2)+1)/x, x ne 0`

Answer

To differentiate the function \( y = \frac{x^2 + 1}{x} \) from first principles (the definition of the derivative), we will use the limit definition of the derivative:

\[ \frac{dy}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]

### Step 1: Define the function
Let \( f(x) = \frac{x^2 + 1}{x} \).

### Step 2: Compute \( f(x + h) \)
We need to find \( f(x + h) \):
\[ f(x + h) = \frac{(x + h)^2 + 1}{x + h} \]
Expanding \( (x + h)^2 \):
\[ f(x + h) = \frac{x^2 + 2xh + h^2 + 1}{x + h} \]

### Step 3: Set up the difference quotient
Now we compute \( f(x + h) - f(x) \):
\[ f(x + h) - f(x) = \frac{x^2 + 2xh + h^2 + 1}{x + h} - \frac{x^2 + 1}{x} \]

### Step 4: Find a common denominator
The common denominator for the two fractions is \( x(x + h) \):
\[ f(x + h) - f(x) = \frac{(x^2 + 2xh + h^2 + 1)x - (x^2 + 1)(x + h)}{x(x + h)} \]

### Step 5: Simplify the numerator
Expanding both parts in the numerator:
1. \( (x^2 + 2xh + h^2 + 1)x = x^3 + 2x^2h + xh^2 + x \)
2. \( (x^2 + 1)(x + h) = x^3 + x^2h + x + h \)

Now, subtract the second from the first:
\[ x^3 + 2x^2h + xh^2 + x - (x^3 + x^2h + x + h) = (2x^2h - x^2h) + xh^2 - h = x^2h + xh^2 - h \]

### Step 6: Factor the numerator
Factoring out \( h \):
\[ h(x^2 + xh - 1) \]

### Step 7: Substitute into the limit
Now we substitute back into the limit:
\[ \frac{dy}{dx} = \lim_{h \to 0} \frac{h(x^2 + xh - 1)}{h(x(x + h))} = \lim_{h \to 0} \frac{x^2 + xh - 1}{x(x + h)} \]

### Step 8: Evaluate the limit as \( h \to 0 \)
As \( h \to 0 \):
\[ \frac{dy}{dx} = \frac{x^2 - 1}{x^2} = 1 - \frac{1}{x^2} \]

### Final Result
Thus, the derivative of the function \( y = \frac{x^2 + 1}{x} \) is:
\[ \frac{dy}{dx} = 1 - \frac{1}{x^2} \]