the greatest value of an implicit function $$\frac{\exp(x)\sin(x) - \exp(y)\sin(y)}{\exp(y)-\exp(x)} \le P$$ where $x \le y$
how would you find the smallest value of $P$ - the greatest value of the LHS implicit function without using a graphical calculator
In addition $x$ is less than or equal to $y$ that means the graph must lie on the LHS of the graph $y=x$

As $y\to x+$, the function approaches the derivative of $-u\sin(\log u)$.
Otherwise, the two partial derivatives $\partial /\partial x$ and $\partial /\partial y$ of $$\exp x\sin x-\exp y\sin y=P(\exp y-\exp x)$$ both hold.
