https://file.aiquickdraw.com/mathbot/file/1726812897655bas2nw8i.jpg

The question asks to find \( f'(9) \) for the given function \( f(x) = \frac{4}{\sqrt{x}} \).

### Step 1: Express \( f(x) \) in a simpler form
The function can be rewritten as:
\[
f(x) = 4x^{-\frac{1}{2}}
\]

### Step 2: Differentiate \( f(x) \)
Now, apply the power rule of differentiation:
\[
f'(x) = 4 \times -\frac{1}{2} x^{-\frac{3}{2}} = -2x^{-\frac{3}{2}}
\]

### Step 3: Find \( f'(9) \)
Substitute \( x = 9 \) into \( f'(x) \):
\[
f'(9) = -2 \times 9^{-\frac{3}{2}} = -2 \times \frac{1}{27} = -\frac{2}{27}
\]

Thus, \( f'(9) = -\frac{2}{27} \), which corresponds to **Answer A**.

Would you like further explanation on the steps, or do you have any specific questions?

Here are 5 related questions to expand your understanding:
1. How do you apply the power rule of differentiation to more complex functions?
2. What is the general process for differentiating functions with fractional exponents?
3. How would you approach the derivative of functions involving roots of different orders?
4. Can you explain the difference between power functions and other types of functions?
5. What are the applications of this kind of differentiation in physics and engineering?

**Tip**: Always simplify the function before differentiating to make the process easier and more accurate.

Find the derivative f'(9) for the given function f(x) = 4 / √x.

Learn how to find the derivative of the function f(x) = 4 / √x and calculate f'(9) using the power rule of differentiation. This problem covers advanced calculus concepts suitable for Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation