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The problem in the image involves differentiating the function \( y = \ln(x + \sqrt{x^2}) \).

### Step-by-Step Solution:
We want to compute \( \frac{dy}{dx} \), the derivative of the function.

The function is:
\[
y = \ln(x + \sqrt{x^2})
\]

#### Step 1: Simplify the expression inside the logarithm
Notice that \( \sqrt{x^2} = |x| \), so the function becomes:
\[
y = \ln(x + |x|)
\]
This simplifies further depending on the sign of \( x \):
- If \( x > 0 \), then \( |x| = x \), and the expression inside the logarithm becomes:
  \[
  y = \ln(x + x) = \ln(2x)
  \]
- If \( x < 0 \), the expression becomes \( \ln(0) \), which is undefined. So the function is only valid for \( x > 0 \).

Thus, for \( x > 0 \), we now have \( y = \ln(2x) \).

#### Step 2: Differentiate \( y = \ln(2x) \)
Using the chain rule:
\[
\frac{dy}{dx} = \frac{1}{2x} \times 2 = \frac{1}{x}
\]

### Final Answer:
For \( x > 0 \), the derivative of the given function is:
\[
\frac{dy}{dx} = \frac{1}{x}
\]

Would you like to explore more details on logarithmic differentiation or work through similar examples?

### Related Questions:
1. How do you differentiate \( \ln(x^2 + 1) \)?
2. What is the derivative of \( \ln(x + \sqrt{x^2 + 4}) \)?
3. Can you differentiate \( \ln(f(x)) \) where \( f(x) \) is a more complex function?
4. How does the derivative of natural logarithms apply in practical problems?
5. Why does \( \sqrt{x^2} = |x| \) instead of just \( x \)?

**Tip:** When simplifying logarithmic functions before differentiation, always check if the argument can be further reduced or simplified for easier calculation.

Prove that the derivative of y = ln(x + sqrt(x^2)) is dy/dx = 1/sqrt(x^2 - 1).

This detailed guide walks through the differentiation of y = ln(x + sqrt(x^2)) using logarithmic properties and chain rule. Learn how to simplify expressions and apply derivatives correctly. Suitable for advanced high school or university-level calculus.

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