https://file.aiquickdraw.com/mathbot/file/1727718056045flb54wma.jpg

The image you provided contains the function \( f(x) = \frac{5}{\sqrt{9 - 2x}} \). We are tasked with finding \( f'(a) \), the derivative of \( f(x) \) evaluated at \( x = a \).

### Steps to find the derivative:

1. **Rewrite the function:**  
   We can rewrite \( f(x) = 5 \cdot (9 - 2x)^{-\frac{1}{2}} \) to make differentiation easier.

2. **Differentiate the function:**  
   Using the chain rule, the derivative of \( f(x) \) is:
   \[
   f'(x) = 5 \cdot \left( -\frac{1}{2} \right) \cdot (9 - 2x)^{-\frac{3}{2}} \cdot (-2)
   \]
   Simplifying:
   \[
   f'(x) = \frac{5}{(9 - 2x)^{\frac{3}{2}}}
   \]

3. **Evaluate at \( x = a \):**  
   Now, substitute \( x = a \) to get \( f'(a) \):
   \[
   f'(a) = \frac{5}{(9 - 2a)^{\frac{3}{2}}}
   \]

So, \( f'(a) = \frac{5}{(9 - 2a)^{\frac{3}{2}}} \).

Would you like further clarification or details?

### Related Questions:
1. What is the general process for applying the chain rule in differentiation?
2. How would you find the second derivative \( f''(x) \) for this function?
3. Can the function \( f(x) \) have any points of discontinuity?
4. How would the derivative change if the square root function was replaced by a different exponent?
5. How does this derivative behave as \( x \) approaches certain critical values?

**Tip:** Always remember to simplify functions, like rewriting square roots as fractional exponents, before differentiating!

Learn how to differentiate the function f(x) = 5 / sqrt(9 - 2x) using the chain rule. This problem walks through rewriting the function for easier differentiation and applying the chain rule to find the derivative. Suitable for advanced high school or early college students.

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