1. The problem is to find the derivative of the function $y = e^{x^x}$ with respect to $x$. 2. We use the chain rule for differentiation, which states that if $y = e^u$ where $u$ is a function of $x$, then $\frac{dy}{dx} = e^u \cdot \frac{du}{dx}$. 3. Here, $u = x^x$. To differentiate $u$, we use logarithmic differentiation: $$u = x^x \implies \ln u = x \ln x$$ 4. Differentiate both sides with respect to $x$: $$\frac{1}{u} \frac{du}{dx} = \ln x + 1$$ 5. Multiply both sides by $u$ to solve for $\frac{du}{dx}$: $$\frac{du}{dx} = x^x (\ln x + 1)$$ 6. Substitute back into the derivative of $y$: $$\frac{dy}{dx} = e^{x^x} \cdot x^x (\ln x + 1)$$ 7. Therefore, the derivative is: $$\boxed{\frac{dy}{dx} = e^{x^x} \cdot x^x (\ln x + 1)}$$