Differentiate the following with respect to x using first principle method.
`(2x+3)/(x+1)`

Answer

To differentiate the function \( f(x) = \frac{2x + 3}{x + 1} \) using the first principle method, we will follow these steps:

### Step 1: Define the function
Let \( f(x) = \frac{2x + 3}{x + 1} \).

### Step 2: Write the formula for the derivative using the first principle
The derivative \( f'(x) \) can be defined using the limit:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \]

### Step 3: Calculate \( f(x + h) \)
Substituting \( x + h \) into the function:
\[ f(x + h) = \frac{2(x + h) + 3}{(x + h) + 1} = \frac{2x + 2h + 3}{x + h + 1} \]

### Step 4: Substitute \( f(x + h) \) and \( f(x) \) into the derivative formula
Now we substitute \( f(x + h) \) and \( f(x) \) into the limit:
\[ f'(x) = \lim_{h \to 0} \frac{\frac{2x + 2h + 3}{x + h + 1} - \frac{2x + 3}{x + 1}}{h} \]

### Step 5: Simplify the expression
To simplify, we need a common denominator:
\[ f'(x) = \lim_{h \to 0} \frac{(2x + 2h + 3)(x + 1) - (2x + 3)(x + h + 1)}{h \cdot (x + h + 1)(x + 1)} \]

### Step 6: Expand the numerator
Now we expand both terms in the numerator:
1. \( (2x + 2h + 3)(x + 1) = 2x^2 + 2xh + 3x + 2h + 3 \)
2. \( (2x + 3)(x + h + 1) = 2x^2 + 2xh + 3x + 3h + 3 \)

Now, substituting these back into the limit:
\[ f'(x) = \lim_{h \to 0} \frac{(2x^2 + 2xh + 3x + 2h + 3) - (2x^2 + 2xh + 3x + 3h + 3)}{h \cdot (x + h + 1)(x + 1)} \]

### Step 7: Combine like terms
Combining like terms in the numerator:
\[ = \lim_{h \to 0} \frac{(2h - 3h)}{h \cdot (x + h + 1)(x + 1)} = \lim_{h \to 0} \frac{-h}{h \cdot (x + h + 1)(x + 1)} \]

### Step 8: Cancel \( h \)
We can cancel \( h \) from the numerator and denominator:
\[ = \lim_{h \to 0} \frac{-1}{(x + h + 1)(x + 1)} \]

### Step 9: Take the limit as \( h \to 0 \)
Now we can take the limit:
\[ = \frac{-1}{(x + 0 + 1)(x + 1)} = \frac{-1}{(x + 1)^2} \]

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