Differentiate the following with respect to 'x' using first principle : `(ax+b)/(cx+d)`

Answer

To differentiate the function \( f(x) = \frac{ax + b}{cx + d} \) using the first principle of derivatives, we follow these steps:

### Step 1: Define the function
Let:
\[ f(x) = \frac{ax + b}{cx + d} \]

### Step 2: Apply the first principle of derivatives
According to the first principle, the derivative \( f'(x) \) is given by:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \]

### Step 3: Calculate \( f(x + h) \)
We need to find \( f(x + h) \):
\[ f(x + h) = \frac{a(x + h) + b}{c(x + h) + d} = \frac{ax + ah + b}{cx + ch + d} \]

### Step 4: Substitute \( f(x + h) \) and \( f(x) \) into the derivative formula
Now, substitute \( f(x + h) \) and \( f(x) \) into the limit:
\[ f'(x) = \lim_{h \to 0} \frac{\frac{ax + ah + b}{cx + ch + d} - \frac{ax + b}{cx + d}}{h} \]

### Step 5: Find a common denominator
To simplify the expression, we find a common denominator:
\[ f'(x) = \lim_{h \to 0} \frac{(ax + ah + b)(cx + d) - (ax + b)(cx + ch + d)}{h \cdot (cx + ch + d)(cx + d)} \]

### Step 6: Expand the numerator
Expanding the numerator:
1. For \( (ax + ah + b)(cx + d) \):
\[ = acx^2 + adh + bcx + bd \]
2. For \( (ax + b)(cx + ch + d) \):
\[ = acx^2 + achx + bcx + bd \]

Now, subtract the second expansion from the first:
\[ = (acx^2 + adh + bcx + bd) - (acx^2 + achx + bcx + bd) \]
This simplifies to:
\[ = adh - achx \]

### Step 7: Factor out common terms
Factoring out \( h \) from the numerator:
\[ = h(ad - acx) \]

### Step 8: Substitute back into the limit
Now substitute back into the limit:
\[ f'(x) = \lim_{h \to 0} \frac{h(ad - acx)}{h \cdot (cx + ch + d)(cx + d)} \]
Cancel \( h \):
\[ f'(x) = \lim_{h \to 0} \frac{ad - acx}{(cx + ch + d)(cx + d)} \]

### Step 9: Evaluate the limit as \( h \to 0 \)
As \( h \to 0 \), \( cx + ch + d \) approaches \( cx + d \):
\[ f'(x) = \frac{ad - acx}{(cx + d)^2} \]

### Final Result
Thus, the derivative of \( f(x) = \frac{ax + b}{cx + d} \) is:
\[ f'(x) = \frac{ad - bc}{(cx + d)^2} \]

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