1. **State the problem:** Find the derivative of the function $f(x) = e^x \ln x$. 2. **Recall the formula:** To differentiate a product of two functions, use the product rule: $$\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$$ where $u(x) = e^x$ and $v(x) = \ln x$. 3. **Differentiate each part:** - Derivative of $u(x) = e^x$ is $u'(x) = e^x$. - Derivative of $v(x) = \ln x$ is $v'(x) = \frac{1}{x}$. 4. **Apply the product rule:** $$f'(x) = e^x \ln x + e^x \cdot \frac{1}{x}$$ 5. **Simplify the expression:** $$f'(x) = e^x \ln x + \frac{e^x}{x} = e^x \left(\ln x + \frac{1}{x}\right)$$ **Final answer:** $$\boxed{f'(x) = e^x \left(\ln x + \frac{1}{x}\right)}$$