Partial derivatives acting on each other

Question

I'm stuck following a step done over and over again in my thermodynamics lecture.

$\frac{\partial^2}{\partial T \partial \beta} = \frac{\partial}{\partial T} \frac{\partial}{\partial (1/kT)} = \frac{1}{kT^2}\frac{\partial^2}{\partial^2(1/kT)} = k\beta^2\frac{\partial^2}{\partial\beta^2}$ and for readability the inverse temperature $\beta=1/kT$

It seems simple enough, but I am not even able to verify that formula. What kind of rules apply here?

Answer 1

Hint

$$\frac{\partial^2A}{\partial T \partial \beta} = \frac{\partial}{\partial T} \Big(\frac{\partial A}{\partial \beta}\Big) = \frac{\partial}{\partial \beta} \Big(\frac{\partial A}{\partial \beta}\Big)\frac{\partial \beta}{\partial T}=\frac{\partial^2A}{ \partial \beta^2}\times \frac{-1}{k T^2}=-k \beta^2\frac{\partial^2A}{ \partial \beta^2}$$