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The given problem asks to find the derivative \( \frac{dy}{dx} \) of the function \( y = \ln(x^2 - 1) \).

### Step-by-Step Solution:

Given:  
\( y = \ln(x^2 - 1) \)

We need to find \( \frac{dy}{dx} \).

#### Step 1: Apply the Chain Rule
The derivative of \( \ln(u) \) with respect to \( u \) is \( \frac{1}{u} \), and we must multiply by the derivative of \( u \) with respect to \( x \).

Here, \( u = x^2 - 1 \), so:

\[
\frac{dy}{dx} = \frac{1}{x^2 - 1} \times \frac{d}{dx}(x^2 - 1)
\]

#### Step 2: Differentiate the Inner Function
Now, differentiate \( x^2 - 1 \) with respect to \( x \):

\[
\frac{d}{dx}(x^2 - 1) = 2x
\]

#### Step 3: Combine the Results
Now, substitute the derivative of the inner function into the chain rule expression:

\[
\frac{dy}{dx} = \frac{2x}{x^2 - 1}
\]

### Final Answer:
The derivative is \( \frac{2x}{x^2 - 1} \), which corresponds to option \( \textbf{B} \) in the provided choices.

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Do you need further details or have any questions?

Here are 5 related questions:
1. What is the chain rule and how is it used in differentiation?
2. How would the derivative change if \( y = \ln(x^2 + 1) \)?
3. What is the significance of logarithmic differentiation?
4. How does the derivative of a logarithmic function differ from that of a power function?
5. What are some real-world applications of logarithmic derivatives?

**Tip:** Always check for functions within functions (composite functions) when differentiating, as this signals the need for the chain rule.

Find the derivative \( \frac{dy}{dx} \) of the function \( y = \ln(x^2 - 1) \).

Explore the process of differentiating the logarithmic function \( y = \ln(x^2 - 1) \) using the chain rule. This guide provides a step-by-step solution that details how to apply calculus concepts to find derivatives effectively.

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