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To solve \( \frac{dy}{dx} \) using logarithmic differentiation for the given function:

\[
y = \sqrt{\frac{x^2 - 16}{x^2 + 16}}, \quad x > 4,
\]

we will follow these steps:

1. Take the natural logarithm of both sides:
   \[
   \ln y = \frac{1}{2} \ln \left( \frac{x^2 - 16}{x^2 + 16} \right).
   \]

2. Simplify using logarithmic properties:
   \[
   \ln y = \frac{1}{2} \left[ \ln(x^2 - 16) - \ln(x^2 + 16) \right].
   \]

3. Differentiate both sides with respect to \(x\):
   \[
   \frac{1}{y} \frac{dy}{dx} = \frac{1}{2} \left[ \frac{1}{x^2 - 16} \cdot 2x - \frac{1}{x^2 + 16} \cdot 2x \right].
   \]

4. Simplify the derivative:
   \[
   \frac{1}{y} \frac{dy}{dx} = \frac{x}{x^2 - 16} - \frac{x}{x^2 + 16}.
   \]

5. Combine the terms under a common denominator:
   \[
   \frac{1}{y} \frac{dy}{dx} = \frac{x \left[ (x^2 + 16) - (x^2 - 16) \right]}{(x^2 - 16)(x^2 + 16)}.
   \]

6. Simplify the numerator:
   \[
   \frac{1}{y} \frac{dy}{dx} = \frac{x(32)}{(x^2 - 16)(x^2 + 16)}.
   \]

7. Multiply by \(y\) to solve for \(\frac{dy}{dx}\):
   \[
   \frac{dy}{dx} = y \cdot \frac{32x}{(x^2 - 16)(x^2 + 16)}.
   \]

8. Substitute \(y = \sqrt{\frac{x^2 - 16}{x^2 + 16}}\):
   \[
   \frac{dy}{dx} = \sqrt{\frac{x^2 - 16}{x^2 + 16}} \cdot \frac{32x}{(x^2 - 16)(x^2 + 16)}.
   \]

9. Simplify the final result:
   \[
   \frac{dy}{dx} = \frac{32x \sqrt{\frac{x^2 - 16}{x^2 + 16}}}{(x^2 - 16)(x^2 + 16)}.
   \]

Let me know if you want additional clarification or to delve deeper into the steps!

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**Relative Questions:**

1. What are the conditions under which logarithmic differentiation is most useful?
2. How does the chain rule factor into logarithmic differentiation?
3. What are the key properties of logarithms that simplify differentiation?
4. Why does the domain \(x > 4\) matter in this problem?
5. How does this solution compare to directly differentiating without logarithms?

**Tip:** When applying logarithmic differentiation, always check for simplifications in the logarithmic expression before differentiating!

Use logarithmic differentiation to find dy/dx for y = sqrt((x^2 - 16) / (x^2 + 16)), x > 4.

Find the derivative of y = sqrt((x^2 - 16) / (x^2 + 16)) using logarithmic differentiation. This step-by-step solution covers properties of logarithms and the chain rule, suitable for advanced calculus students.

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