Is the order of partial differentiation always immaterial for mixed derivatives?

[Diophantus]

I am familiar with the standard rule of mixed partial derivatives in that the order in which you partially differentiate dosn't matter. I have just been considering whether the same rule applies if we take f(q,t) say where q=q(t) and we differentiate normally w.r.t t then partially w.r.t q. Is the order of these operations always immaterial in this case too? I can't find a counterexample but I havn't yet got a satisfactory insight into this problem.

Anyone want to enlighten me?

 

[Galileo]

As long as $$\frac{\partial^2 f}{\partial q \partial t}=\frac{\partial^2 f}{\partial t \partial q}$$ it doesn't matter. All derivatives must exist and be continuous ofcourse.

You could apply the multivariate chain-rule to show it explicitly.

 

[Diophantus]

Maybe I wasn't quite clear enough. I was wodering if:
$$\frac{\partial^2 f}{\partial qdt \dt}=\frac{\partial^2 f}{\d dt \partial q}$$
always holds for f(q,t).

 

[TD]

If the these mixed derivatives exist and are continuous, then they're the same.