Mathematics

This problem asks you to find a derivative when $y$ is defined implicitly through a transcendental equation rather than as an explicit function of $x$. The key insight is to differentiate both sides of the equation $y \ln y = x$ with respect to $x$ using the product rule on the left side, treating $y$ as a function of $x$. Once you have $\frac{dy}{dx}$ in terms of $y$, substituting $y = e$ gives you the specific slope at that point.

This problem asks you to find a derivative when $y$ is defined implicitly through a transcendental equation rather than as an explicit function of $x$. The key insight is to differentiate both sides of the equation $y \ln y = x$ with respect to $x$ using the product rule on the left side, treating $y$ as a function of $x$. Once you have $\frac{dy}{dx}$ in terms of $y$, substituting $y = e$ gives you the specific slope at that point.