https://file.aiquickdraw.com/mathbot/file/1726309544656mf5aeak3.jpg

The OCR extraction was unclear, but based on the image and its structure, the question appears to be:

"Using the Chain Rule, find the value of \( \frac{dy}{dx} \) at the point \( x = 0 \), where:

\[
y = \sqrt{64 + 512 \tan(8x)}
\]

Options:
- 135
- 256
- 106
- 3.931"

Let's work through this step-by-step to find the derivative \( \frac{dy}{dx} \) at \( x = 0 \).

### Step-by-Step Solution:

1. **Identify the function:**
   \[
   y = \sqrt{64 + 512 \tan(8x)}
   \]

2. **Use the Chain Rule:**
   - The outer function is \( y = \sqrt{u} \), where \( u = 64 + 512 \tan(8x) \).
   - The derivative of \( \sqrt{u} \) is \( \frac{1}{2\sqrt{u}} \).
   
   Next, we find the derivative of the inner function \( u = 64 + 512 \tan(8x) \).
   
3. **Differentiate the inner function:**
   \[
   \frac{du}{dx} = 512 \cdot \frac{d}{dx}[\tan(8x)]
   \]
   The derivative of \( \tan(8x) \) is \( 8 \sec^2(8x) \), so:
   \[
   \frac{du}{dx} = 512 \cdot 8 \sec^2(8x) = 4096 \sec^2(8x)
   \]

4. **Apply the Chain Rule:**
   Using the Chain Rule, \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \):
   \[
   \frac{dy}{dx} = \frac{1}{2\sqrt{64 + 512 \tan(8x)}} \cdot 4096 \sec^2(8x)
   \]

5. **Evaluate at \( x = 0 \):**
   At \( x = 0 \):
   - \( \tan(8 \cdot 0) = 0 \), so the expression inside the square root becomes:
     \[
     64 + 512 \cdot 0 = 64
     \]
   - Therefore, \( \sqrt{64} = 8 \).
   
   Substituting into the derivative:
   \[
   \frac{dy}{dx} = \frac{1}{2 \cdot 8} \cdot 4096 \cdot \sec^2(0)
   \]
   Since \( \sec(0) = 1 \), we get:
   \[
   \frac{dy}{dx} = \frac{1}{16} \cdot 4096 = 256
   \]

Thus, the value of \( \frac{dy}{dx} \) at \( x = 0 \) is **256**.

Would you like further details or have any other questions?

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### Related Questions:
1. How is the chain rule applied when multiple functions are involved?
2. What is the derivative of \( \tan(x) \), and why is it important in trigonometric differentiation?
3. Can you explain how the secant function behaves and its properties?
4. How would this process change if the function were \( y = \sin(x) \) instead of \( \tan(x) \)?
5. What is the geometric interpretation of the derivative at a point?

### Tip:
Always simplify inside the square root or trigonometric functions before differentiating to reduce errors!

Using the Chain Rule, find the value of dy/dx at the point x=0 where y = sqrt(64 + 512*tan(8x))

Find the derivative of y = sqrt(64 + 512*tan(8x)) using the Chain Rule. This calculus problem involves differentiating square root and trigonometric functions and applying the Chain Rule. The solution walks through step-by-step, finding that dy/dx at x=0 is 256. Suitable for college-level calculus students.

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