Differentiate `sqrt( ax+b)` with respect to `x` from definition.

Answer

To differentiate the function \( f(x) = \sqrt{ax + b} \) with respect to \( x \) using the first principle of differentiation, we follow these steps:

### Step 1: Define the function and the increment
We start with the function:
\[ f(x) = \sqrt{ax + b} \]
We also need to consider \( f(x + h) \):
\[ f(x + h) = \sqrt{a(x + h) + b} = \sqrt{ax + ah + b} \]

### Step 2: Apply the definition of the derivative
The derivative \( f'(x) \) using the first principle is given by:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \]
Substituting our expressions for \( f(x + h) \) and \( f(x) \):
\[ f'(x) = \lim_{h \to 0} \frac{\sqrt{ax + ah + b} - \sqrt{ax + b}}{h} \]

### Step 3: Rationalize the numerator
To simplify the expression, we multiply the numerator and denominator by the conjugate of the numerator:
\[ f'(x) = \lim_{h \to 0} \frac{\left(\sqrt{ax + ah + b} - \sqrt{ax + b}\right)\left(\sqrt{ax + ah + b} + \sqrt{ax + b}\right)}{h\left(\sqrt{ax + ah + b} + \sqrt{ax + b}\right)} \]
This simplifies to:
\[ f'(x) = \lim_{h \to 0} \frac{(ax + ah + b) - (ax + b)}{h\left(\sqrt{ax + ah + b} + \sqrt{ax + b}\right)} \]

### Step 4: Simplify the numerator
The numerator simplifies to:
\[ (ax + ah + b) - (ax + b) = ah \]
Thus, we have:
\[ f'(x) = \lim_{h \to 0} \frac{ah}{h\left(\sqrt{ax + ah + b} + \sqrt{ax + b}\right)} \]
Cancelling \( h \) in the numerator and denominator gives:
\[ f'(x) = \lim_{h \to 0} \frac{a}{\sqrt{ax + ah + b} + \sqrt{ax + b}} \]

### Step 5: Evaluate the limit
Now, we can substitute \( h \) with \( 0 \):
\[ f'(x) = \frac{a}{\sqrt{ax + 0 + b} + \sqrt{ax + b}} = \frac{a}{2\sqrt{ax + b}} \]

### Final Answer
Thus, the derivative of \( \sqrt{ax + b} \) with respect to \( x \) is:
\[ f'(x) = \frac{a}{2\sqrt{ax + b}} \]

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