Given the function $y = 3^{x^2}$, we want to find the derivative $\frac{dy}{dx}$.

Steps:

1. Recognize that this is an exponential function where the exponent itself is a function of $x$.

2. Use the chain rule and logarithmic differentiation. For a function of the form $y = a^{u(x)}$, the derivative is: $\frac{dy}{dx} = a^{u(x)} \cdot \ln(a) \cdot \frac{du}{dx}$ Here, $a = 3$ and $u(x) = x^2$.

3. Differentiate $y = 3^{x^2}$ using the formula above: $\frac{dy}{dx} = 3^{x^2} \cdot \ln(3) \cdot \frac{d}{dx}(x^2)$

4. The derivative of $x^2$ is $2x$, so: $\frac{dy}{dx} = 3^{x^2} \cdot \ln(3) \cdot 2x$

Thus, the derivative is: $\frac{dy}{dx} = 2x \cdot 3^{x^2} \cdot \ln(3)$

Would you like a deeper explanation on any step or additional details?

Here are 5 related questions to explore further:

1. How do you apply the chain rule for other functions?
2. How would you differentiate $e^{x^2}$ instead of $3^{x^2}$?
3. How does logarithmic differentiation work in general?
4. How does the derivative of a constant base exponential function differ from a variable base?
5. What happens if we apply implicit differentiation in such problems?

Tip: When differentiating exponential functions, the logarithmic base of the exponent is crucial in getting the right factor for the derivative.