What About Schwarz Theorem?

Consider this,

\(T=T_oe^{-iwt}\)   obviously,  \(T=f(t,T)\)

\(\cfrac{dT}{d\,t}=-iwT_oe^{-iwt}\)

\(\cfrac{d^2T}{d\,T d\,t}=-iw\)

But,

\(\cfrac{d^2T}{d\,Td\,t}=\cfrac{d}{d\,t}\left\{\cfrac{d\,T}{d\,T}\right\}=\cfrac{d}{d\,t}\left\{1\right\}=0\)

Who's Schwarz?