I encountered this in a derivation of the 1D wave equation.
Why does the order of application not matter?
$$ expr = f(x,t)$$
$$ \frac{\partial }{\partial t} \left( \frac{\partial }{\partial x } \left(expr \right ) \right) = \frac{\partial }{\partial x} \left( \frac{\partial }{\partial t } \left(expr \right ) \right) $$
why is this true?
(Is it also true for Total Derivatives in the same situation?)
By Schwarz's/Clairaut's/Young's Theorem $$f_{xt}(x,t)=f_{tx}(x,t)$$ or in your notation
$$\frac{\partial^2 f}{\partial t\partial x}=\frac{\partial^2 f}{\partial x\partial t}$$
if $f$ is twice continuously differentiable. (A weaker sufficient condition for symmetry of second-order partial derivatives is that all first-order partial derivatives are differentiable.)
About your question regarding "total derivatives" if you mean the partial derivatives of a composition of functions then yes, if the composition is twice continuously differentiable.
