 Inverse derivative of an exponential coordinate map

This map is a diffeomorphism from ${\mathbb{R}}^2$ onto $V=\{(s,t):s>0,t>0,st>1\}$. Writing $q=e^x$, the inverse problem becomes $(s-q)(t-q)=1$ with $0<q<\min (s,t)$. The left side decreases strictly from $st$ to zero, proving existence and uniqueness exactly on $V$. Its Jacobian determinant is $-e^x(e^y+e^{-y})\ne 0$, so the inverse function theorem supplies a differentiable inverse. Along $(x,y)=(c,c)$, applying the inverse Jacobian to $(1/2,1/2)$ gives $(e^{-c}/2,0)$. Establishing the actual image matters before using a global inverse, even when a local inverse already exists at the point of interest.