The problem is asking for the derivative of the function:

$y = \left(\log_3 x\right)^2$

To differentiate this function, let's apply the chain rule and the properties of logarithmic differentiation.

Step-by-Step Solution

1. Rewrite the Function in Terms of a Natural Logarithm: Since $\log_3 x = \frac{\ln x}{\ln 3}$, we can rewrite $y$ as: $y = \left(\frac{\ln x}{\ln 3}\right)^2$

2. Differentiate Using the Chain Rule: Let $u = \frac{\ln x}{\ln 3}$, so $y = u^2$. The derivative of $y$ with respect to $x$ is: $\frac{dy}{dx} = 2u \cdot \frac{du}{dx}$

3. Differentiate $u$ with Respect to $x$: Since $u = \frac{\ln x}{\ln 3}$, we have: $\frac{du}{dx} = \frac{1}{x \ln 3}$

4. Combine the Results: Substitute $u = \frac{\ln x}{\ln 3}$ and $\frac{du}{dx} = \frac{1}{x \ln 3}$ back into the expression: $\frac{dy}{dx} = 2 \cdot \frac{\ln x}{\ln 3} \cdot \frac{1}{x \ln 3}$

5. Simplify: $\frac{dy}{dx} = \frac{2 \ln x}{x (\ln 3)^2}$

Final Answer

$\frac{dy}{dx} = \frac{2 \ln x}{x (\ln 3)^2}$

Would you like further clarification on any of these steps?

Here are some follow-up questions for deeper understanding:

1. How does the chain rule apply in this problem?
2. Can we generalize this approach for other bases of logarithms?
3. What would happen if the power was different, say $3$ instead of $2$?
4. How does this differentiation change if the argument of the logarithm, $x$, was a more complex function?
5. Why is it useful to rewrite logarithms in terms of natural logs?

Tip: Remember that converting logarithms to natural logarithms can simplify differentiation with unusual bases.