find the derivative analytically of f(x)=sqr(x-3)

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.

Learn how to find the derivative of f(x) = sqrt(x - 3) analytically using the chain rule. This step-by-step solution covers key calculus concepts like the chain rule and the power rule, ideal for students in Grades 11-12 or college calculus courses.

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