1. **Problem:** Find the derivative of $y = (3x^2 + 1)^{3/2}$. 2. **Formula:** Use the chain rule for derivatives: $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$. 3. **Step 1:** Let $u = 3x^2 + 1$, so $y = u^{3/2}$. 4. **Step 2:** Derivative of $u^{3/2}$ with respect to $u$ is $\frac{3}{2} u^{1/2}$. 5. **Step 3:** Derivative of $u = 3x^2 + 1$ with respect to $x$ is $6x$. 6. **Step 4:** Apply chain rule: $$\frac{dy}{dx} = \frac{3}{2} (3x^2 + 1)^{1/2} \cdot 6x = 9x (3x^2 + 1)^{1/2}$$ 7. **Answer:** The derivative is $\boxed{9x (3x^2 + 1)^{1/2}}$.