Find the derivative of `sqrt(4-x)` w.r.t. `x` using the first principle.

Explanation

To find the derivative of \( f(x) = \sqrt{4 - x} \) with respect to \( x \) using the first principle of derivatives, we will follow these steps:

### Step 1: Write the definition of the derivative using the first principle
The derivative of a function \( f(x) \) at a point \( x \) is defined as:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \]

### Step 2: Identify \( f(x) \) and \( f(x + h) \)
Given \( f(x) = \sqrt{4 - x} \), we need to find \( f(x + h) \):
\[ f(x + h) = \sqrt{4 - (x + h)} = \sqrt{4 - x - h} \]

### Step 3: Substitute \( f(x) \) and \( f(x + h) \) into the derivative formula
Now, substituting into the derivative formula:
\[ f'(x) = \lim_{h \to 0} \frac{\sqrt{4 - x - h} - \sqrt{4 - x}}{h} \]

### Step 4: Rationalize the numerator
To simplify the expression, we multiply and divide by the conjugate of the numerator:
\[ f'(x) = \lim_{h \to 0} \frac{\left(\sqrt{4 - x - h} - \sqrt{4 - x}\right)\left(\sqrt{4 - x - h} + \sqrt{4 - x}\right)}{h \left(\sqrt{4 - x - h} + \sqrt{4 - x}\right)} \]
This simplifies the numerator using the difference of squares:
\[ = \lim_{h \to 0} \frac{(4 - x - h) - (4 - x)}{h \left(\sqrt{4 - x - h} + \sqrt{4 - x}\right)} \]

### Step 5: Simplify the numerator
The numerator simplifies to:
\[ = \lim_{h \to 0} \frac{-h}{h \left(\sqrt{4 - x - h} + \sqrt{4 - x}\right)} \]
Now, we can cancel \( h \) in the numerator and denominator:
\[ = \lim_{h \to 0} \frac{-1}{\sqrt{4 - x - h} + \sqrt{4 - x}} \]

### Step 6: Evaluate the limit as \( h \to 0 \)
As \( h \) approaches 0, \( \sqrt{4 - x - h} \) approaches \( \sqrt{4 - x} \):
\[ = \frac{-1}{\sqrt{4 - x} + \sqrt{4 - x}} = \frac{-1}{2\sqrt{4 - x}} \]

### Final Result
Thus, the derivative of \( f(x) = \sqrt{4 - x} \) is:
\[ f'(x) = -\frac{1}{2\sqrt{4 - x}} \]

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