1. **State the problem:** We want to understand the function $y = x^x$. 2. **Formula and explanation:** The function $y = x^x$ is defined for $x > 0$ because the base and the exponent are both $x$. This is an example of a function where the variable appears both as the base and the exponent. 3. **Rewrite the function:** We can rewrite $x^x$ using the exponential and logarithm functions: $$x^x = e^{x \ln x}$$ This helps us analyze and differentiate the function. 4. **Domain:** Since $\ln x$ is defined only for $x > 0$, the domain of $y = x^x$ is $x > 0$. 5. **Derivative (optional for deeper understanding):** Using the rewritten form, the derivative is: $$\frac{dy}{dx} = \frac{d}{dx} e^{x \ln x} = e^{x \ln x} \cdot \frac{d}{dx} (x \ln x)$$ 6. **Simplify the derivative:** $$\frac{d}{dx} (x \ln x) = \ln x + 1$$ 7. **Final derivative:** $$\frac{dy}{dx} = x^x (\ln x + 1)$$ This derivative tells us how the function changes with respect to $x$. **Final answer:** The function is $y = x^x$ for $x > 0$.