$$\frac{\exp(x)\sin(x) - \exp(y)\sin(y)}{\exp(y)-\exp(x)} \le P$$ 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 $y=x$
On
By setting $u=\exp x, v=\exp y$ and $f(w)=-w\sin\log w$ we are looking for $$P=\sup_{0<u<v}\frac{f(v)-f(u)}{v-u}$$ but the Lagrange theorem gives: $$\frac{f(v)-f(u)}{v-u}=f'(\xi),\qquad \xi\in(u,v)$$ and since: $$ f'(w) = -\sin(\log w)-\cos(\log w)=-\sqrt{2}\sin(\log w+\pi/4)$$ we have $\color{red}{P=\sqrt{2}}$.
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.