Can we rearrage the second derivate doing this:

Question

There is a homogeneous of degree 1 function $I=f(T,M)$, we find the first derivate: $$ \frac{\partial I}{\partial M} $$ Next, we take the derivate with respect to T: $$ \frac{\partial I}{\partial M}\frac{\partial}{\partial T} $$ Is this a true statement? $$ \frac{\partial I}{\partial M}\frac{\partial}{\partial T}=\frac{\partial I}{\partial T}\frac{\partial}{\partial M} $$