For an infinitely differentiable function $f(x,t)$ and $a \in \mathbb{R}$ does the following hold?
$$\frac{\partial}{\partial{t}} \exp\left( \frac{\partial{f(x,a)}}{\partial{x}} \right) = \exp \left( \frac{\partial}{\partial{x}} \frac{\partial{f(x,a)}}{\partial{t}} \right)$$
Well, I have no proof, but I think it is since we can put the limit inside the argument of the exponential function and then use Clairaut's Theorem. Is it correct?
EDIT Let me link with this one: About Partial Derivatives of a function