To find the derivative of the function $f(x) = \sqrt{x - 3}$ analytically, we can follow these steps:

1. Rewrite the function:
   Express $\sqrt{x - 3}$ as a power of 1/2:
   $f(x) = (x - 3)^{1/2}$

2. Apply the Chain Rule:
   The Chain Rule states that if $f(x) = (g(x))^n$, then the derivative is:
   $\frac{d}{dx} f(x) = n \cdot (g(x))^{n-1} \cdot \frac{d}{dx} g(x)$ Here, $g(x) = x - 3$ and $n = 1/2$.

3. Differentiate: Using the Chain Rule: $\frac{d}{dx} \left( (x - 3)^{1/2} \right) = \frac{1}{2} (x - 3)^{-1/2} \cdot \frac{d}{dx}(x - 3)$ The derivative of $x - 3$ with respect to $x$ is 1.

4. Final result:
   So, the derivative is: $f'(x) = \frac{1}{2} (x - 3)^{-1/2}$ This can also be written as: $f'(x) = \frac{1}{2\sqrt{x - 3}}$

This is the derivative of $f(x) = \sqrt{x - 3}$.

Would you like further details or clarifications?

Here are five related questions:

1. What is the derivative of $\sqrt{2x + 1}$?
2. How would you apply the Chain Rule to $f(x) = (x^2 - 1)^{3/2}$?
3. What is the second derivative of $\sqrt{x - 3}$?
4. How do we handle finding the derivative of a nested function?
5. What is the domain of $f'(x) = \frac{1}{2\sqrt{x - 3}}$?

Tip: When working with square roots, it’s helpful to rewrite them as fractional exponents to simplify differentiation.