1. Problem: Find $\frac{dy}{dx}$ for the exponential function $y = e^{3x^2}$. 2. Formula: The derivative of $e^{u(x)}$ with respect to $x$ is given by the chain rule: $$\frac{dy}{dx} = e^{u(x)} \cdot \frac{du}{dx}$$ where $u(x)$ is a differentiable function of $x$. 3. Applying the formula: Here, $u(x) = 3x^2$. Calculate $\frac{du}{dx}$: $$\frac{du}{dx} = \frac{d}{dx}(3x^2) = 6x$$ 4. Substitute back into the derivative formula: $$\frac{dy}{dx} = e^{3x^2} \cdot 6x = 6x e^{3x^2}$$ 5. Explanation: We used the chain rule because the exponent is a function of $x$. First, differentiate the exponent $3x^2$ to get $6x$, then multiply by the original exponential function. Final answer: $$\boxed{\frac{dy}{dx} = 6x e^{3x^2}}$$