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.

Discover how to differentiate the function y = 3^x^2 by applying the chain rule and logarithmic differentiation. This solution includes detailed steps, key formulas, and concepts like exponential functions and the chain rule. Perfect for students in Grades 11-12.

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