Find, from first principles, the derivative of the following w.r.t. x :
`(x^(2)+2)/(x+2)`

Answer

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

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

### Step 1: Write down \( f(x + h) \)

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

\[ f(x + h) = \frac{(x + h)^2 + 2}{(x + h) + 2} \]

Expanding this, we get:

\[ f(x + h) = \frac{x^2 + 2xh + h^2 + 2}{x + h + 2} \]

### Step 2: Calculate \( f(x + h) - f(x) \)

Next, we need to find \( f(x + h) - f(x) \):

\[ f(x + h) - f(x) = \frac{x^2 + 2xh + h^2 + 2}{x + h + 2} - \frac{x^2 + 2}{x + 2} \]

### Step 3: Find a common denominator

The common denominator for the two fractions is \( (x + h + 2)(x + 2) \). Therefore, we can rewrite the expression:

\[ f(x + h) - f(x) = \frac{(x^2 + 2xh + h^2 + 2)(x + 2) - (x^2 + 2)(x + h + 2)}{(x + h + 2)(x + 2)} \]

### Step 4: Simplify the numerator

Now, we will simplify the numerator:

1. Expand \( (x^2 + 2xh + h^2 + 2)(x + 2) \):
\[ = x^3 + 2x^2 + 2xh + 2h^2 + 2x + 4 \]

2. Expand \( (x^2 + 2)(x + h + 2) \):
\[ = x^3 + x^2h + 2x^2 + 2h + 4 \]

3. Combine these expansions:
\[ \text{Numerator} = (x^3 + 2x^2 + 2xh + 2h^2 + 2x + 4) - (x^3 + x^2h + 2x^2 + 2h + 4) \]

4. This simplifies to:
\[ = (2xh + 2h^2 - x^2h - 2h) \]

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

Factoring \( h \) from the numerator gives us:

\[ = h(2x + 2h - x^2 - 2) \]

### Step 6: Substitute back into the limit

Now substituting back into the limit:

\[ f'(x) = \lim_{h \to 0} \frac{h(2x + 2h - x^2 - 2)}{h(x + h + 2)(x + 2)} \]

### Step 7: Cancel \( h \)

We can cancel \( h \) from the numerator and denominator:

\[ f'(x) = \lim_{h \to 0} \frac{2x + 2h - x^2 - 2}{(x + h + 2)(x + 2)} \]

### Step 8: Apply the limit

Now, applying the limit as \( h \to 0 \):

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

### Final Result

Thus, the derivative of the function \( f(x) = \frac{x^2 + 2}{x + 2} \) is:

\[ f'(x) = \frac{x^2 + 4x - 2}{(x + 2)^2} \]