Find the derivative of `(2x + 3)/(4x + 1)` using first principle of derivatives

Answer

To find the derivative of the function \( f(x) = \frac{2x + 3}{4x + 1} \) using the first principle of derivatives, we will follow these steps:

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

### Step 2: Substitute the function into the formula
We have \( f(x) = \frac{2x + 3}{4x + 1} \). Now, we need to find \( f(x + h) \):
\[ f(x + h) = \frac{2(x + h) + 3}{4(x + h) + 1} = \frac{2x + 2h + 3}{4x + 4h + 1} \]

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

### Step 4: Simplify the expression
To simplify the numerator, we need a common denominator:
\[ \text{Numerator} = \frac{(2x + 2h + 3)(4x + 1) - (2x + 3)(4x + 4h + 1)}{(4x + 4h + 1)(4x + 1)} \]

### Step 5: Expand the numerator
Expanding both terms in the numerator:
1. \( (2x + 2h + 3)(4x + 1) = 8x^2 + 2hx + 2h + 12x + 3 \)
2. \( (2x + 3)(4x + 4h + 1) = 8x^2 + 8hx + 2x + 12h + 3 \)

Now, subtract these two results:
\[ \text{Numerator} = (8x^2 + 2hx + 2h + 12x + 3) - (8x^2 + 8hx + 2x + 12h + 3) \]
This simplifies to:
\[ = -6hx + 10h \]

### Step 6: Substitute back into the limit
Now we have:
\[ f'(x) = \lim_{h \to 0} \frac{-6hx + 10h}{h(4x + 4h + 1)(4x + 1)} \]
We can cancel \( h \) from the numerator and denominator:
\[ = \lim_{h \to 0} \frac{-6x + 10}{(4x + 4h + 1)(4x + 1)} \]

### Step 7: Evaluate the limit as \( h \to 0 \)
Substituting \( h = 0 \):
\[ f'(x) = \frac{-6x + 10}{(4x + 1)(4x + 1)} = \frac{-6x + 10}{(4x + 1)^2} \]

### Final Result
Thus, the derivative of the function \( f(x) = \frac{2x + 3}{4x + 1} \) is:
\[ f'(x) = \frac{-6x + 10}{(4x + 1)^2} \]