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?