1. **State the problem:** We need to find the derivative of the function $f(x) = \ln\left( (e^x)^x \right)$. 2. **Simplify the function:** Recall the power rule for exponents: $(a^b)^c = a^{bc}$. So, $$ (e^x)^x = e^{x \cdot x} = e^{x^2} $$ Thus, $$ f(x) = \ln(e^{x^2}) $$ 3. **Use the logarithm property:** The natural logarithm and exponential functions are inverses, so $$ \ln(e^{x^2}) = x^2 $$ 4. **Differentiate the simplified function:** The derivative of $x^2$ with respect to $x$ is $$ \frac{d}{dx} x^2 = 2x $$ 5. **Final answer:** $$ f'(x) = 2x $$