The task is to calculate the derivative of $(\ln^2 x)'$ and select the correct option from the provided choices.

---

Step-by-Step Solution:

1. Rewrite the Expression
   The given expression is $\ln^2 x$, which means:
   $(\ln x)^2$

2. Apply the Chain Rule
   Using the chain rule for derivatives, we have:
   $\frac{d}{dx} [(\ln x)^2] = 2 (\ln x) \cdot \frac{d}{dx}(\ln x)$

3. Derivative of $\ln x$
   The derivative of $\ln x$ is:
   $\frac{d}{dx} (\ln x) = \frac{1}{x}$

4. Substitute
   Substituting back, we get:
   $\frac{d}{dx} [(\ln x)^2] = 2 (\ln x) \cdot \frac{1}{x} = \frac{2 \ln x}{x}$

---

Final Answer:

The correct option is:
$\boxed{\frac{2 \ln x}{x}}$

---

Would you like more details on any part of this solution? 😊
Here are 5 related questions to deepen your understanding:

1. What is the derivative of $\ln^n x$ for any positive integer $n$?
2. How does the chain rule apply when differentiating composite functions?
3. What is the difference between $(\ln x)^2$ and $\ln(x^2)$?
4. How do you differentiate powers of logarithmic functions like $(\ln x)^3$?
5. How is the product rule related to the chain rule in solving this derivative?

Tip:

Always simplify logarithmic expressions before differentiating—it often saves time and reduces errors.