Many times I come across some new formula being used to work with and/or reduce partial differentials. As kleingordon said, these things are mysteriously not taught anywhere(atleast in physics courses). I can't find any list on the internet, either.
I'm talking about formulae like these:
$$\frac{\mathrm{d}}{\mathrm{d}\alpha}\int f(x,\alpha) \mathrm{d}x=\int\frac{\partial f(x,\alpha)}{\partial \alpha}\mathrm{d}x$$
$$\frac{\partial}{\partial x}\frac{\partial f}{\partial y}=\frac{\partial}{\partial y}\frac{\partial f}{\partial x}$$ (for continuous functions)
I've also seen that you can stuff a derivative inside a PD $$ \frac{\rm d}{\rm dt}\left(\frac{\partial f}{\partial x}\right)=\frac{\partial \dot f}{\partial x}$$ (Note-$\dot f=\frac{\rm df}{\rm dt}$)
There's also a formula that allows one to split a function into a sum of partial derivatives. I think this is the multivariable chain rule.
I'd like a list of such formulae, or links to these lists. Books are also fine, though I'd prfer internet sources.
$\def\p{\partial}$Here's a proof of your last statement. It uses the chain rule: for functions $x(t)$ and $g(x,t)$ you have
$$\frac{d}{dt} g(x,t) = \frac{\p g}{\p t} + \frac{\p g}{\p x} \frac{dx}{dt} \tag{1}$$
If you take $g=\p f/\p x$, then plugging into (1) gives
$$\frac{d}{dt} \frac{\p f}{\p x} = \frac{\p^2 f}{\p t \p x} + \frac{\p^2 f}{\p x^2} \frac{dx}{dt}$$
On the other hand, if first take $g=f$ and then take the partial derivative with respect to $x$:
$$\frac{\p}{\p x} \frac{df}{dt} = \frac{\p}{\p x} \left( \frac{\p f}{\p t} + \frac{\p f}{\p x} \frac{dx}{dt} \right) = \frac{\p^2 f}{\p x\p t} + \frac{\p^2f}{\p x^2} \frac{dx}{dt}$$
You can compare the right-hand sides of these expressions and see that they are equal (since partial derivatives commute). Therefore
$$\frac{d}{dt} \frac{\p f}{\p x} = \frac{\p}{\p x} \frac{df}{dt}$$
so the partial derivative wrt $x$ commutes with the total derivative wrt $t$.