Differentiate from first principles:
8. `x+ (1)/( x)`

Answer

To differentiate the function \( f(x) = x + \frac{1}{x} \) from first principles, we will follow these steps:

### Step 1: Write the definition of the derivative from first principles.

The derivative of a function \( f(x) \) from first principles is given by:

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

### Step 2: Substitute \( f(x) \) and \( f(x+h) \).

First, we calculate \( f(x+h) \):

\[ f(x+h) = (x+h) + \frac{1}{x+h} \]

Now, substitute \( f(x) \) and \( f(x+h) \) into the derivative formula:

\[ f'(x) = \lim_{h \to 0} \frac{(x+h) + \frac{1}{x+h} - \left( x + \frac{1}{x} \right)}{h} \]

### Step 3: Simplify the expression.

Now, simplify the expression in the limit:

\[ f'(x) = \lim_{h \to 0} \frac{(x+h) - x + \frac{1}{x+h} - \frac{1}{x}}{h} \]

This simplifies to:

\[ f'(x) = \lim_{h \to 0} \frac{h + \left( \frac{1}{x+h} - \frac{1}{x} \right)}{h} \]

### Step 4: Combine the fractions.

To combine the fractions \( \frac{1}{x+h} - \frac{1}{x} \), we find a common denominator:

\[ \frac{1}{x+h} - \frac{1}{x} = \frac{x - (x+h)}{x(x+h)} = \frac{-h}{x(x+h)} \]

Substituting this back into our limit gives:

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

### Step 5: Factor out \( h \).

We can factor \( h \) out of the numerator:

\[ f'(x) = \lim_{h \to 0} \left( 1 - \frac{1}{x(x+h)} \right) \]

### Step 6: Evaluate the limit.

Now, we evaluate the limit as \( h \) approaches 0:

\[ f'(x) = 1 - \frac{1}{x^2} \]

### Step 7: Final result.

Thus, the derivative of the function \( f(x) = x + \frac{1}{x} \) is:

\[ f'(x) = 1 - \frac{1}{x^2} \]

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