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 definition of the derivative
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: Calculate \( f(x+h) \)
First, 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: Substitute \( f(x+h) \) and \( f(x) \) into the derivative formula
Now we 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 expression, we need a common denominator for the two fractions in the numerator:
\[ f'(x) = \lim_{h \to 0} \frac{(2x + 2h + 3)(4x + 1) - (2x + 3)(4x + 4h + 1)}{h(4x + 4h + 1)(4x + 1)} \]

### Step 5: Expand the numerator
Now, we expand the numerator:
\[ (2x + 2h + 3)(4x + 1) = 8x^2 + 2hx + 2h + 12x + 3 \]
\[ (2x + 3)(4x + 4h + 1) = 8x^2 + 8xh + 2x + 12x + 12h + 3 \]
Now, substituting these expansions back into the limit gives:
\[ f'(x) = \lim_{h \to 0} \frac{(8x^2 + 2hx + 2h + 12x + 3) - (8x^2 + 8xh + 2x + 12x + 12h + 3)}{h(4x + 4h + 1)(4x + 1)} \]

### Step 6: Combine like terms in the numerator
Combining like terms in the numerator:
\[ = \lim_{h \to 0} \frac{-6xh - 10h}{h(4x + 4h + 1)(4x + 1)} \]

### Step 7: Factor out \( h \) from the numerator
Factoring out \( h \):
\[ = \lim_{h \to 0} \frac{h(-6x - 10)}{h(4x + 4h + 1)(4x + 1)} \]
Canceling \( h \):
\[ = \lim_{h \to 0} \frac{-6x - 10}{(4x + 4h + 1)(4x + 1)} \]

### Step 8: Evaluate the limit as \( h \to 0 \)
Now we can evaluate the limit:
\[ = \frac{-6x - 10}{(4x + 1)(4x + 1)} = \frac{-6x - 10}{(4x + 1)^2} \]

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