Derivative *of* a partial derivative

[Pollywoggy]

Homework Statement

In books I have been using to learn about the Lagrangian function, I find equations that have a derivative of a partial derivative, as in the snippet below. Is there a place where I can learn how this works and *why* it works? I think I can do it mechanically but I want to understand what I am doing. What type of calculus books would have an explanation of it, advanced calculus books?

Homework Equations

\frac{d}{dt} (\displaystyle \frac{\partial T}{\partial \dot{q}_\alpha})

 

[Clever-Name]

What do you mean HOW does it work... You're just taking the derivative of something that has already been derivated.

It's like

F(x,t) = xe^{t^{2}}

\frac{\partial F}{\partial x} = e^{t^{2}}

\frac{d}{dt}(\frac{\partial F}{\partial x}) = 2te^{t^{2}}

To me it's fairly straightforward.. You're just taking a derivative wrt time of a derivative wrt something else. I don't understand what you mean when you ask 'WHY' this works.

 

[Pollywoggy]

What do you mean HOW does it work... You're just taking the derivative of something that has already been derivated.

It's like

F(x,t) = xe^{t^{2}}

\frac{\partial F}{\partial x} = e^{t^{2}}

\frac{d}{dt}(\frac{\partial F}{\partial x}) = 2te^{t^{2}}

To me it's fairly straightforward.. You're just taking a derivative wrt time of a derivative wrt something else. I don't understand what you mean when you ask 'WHY' this works.

Thanks, I think you just answered my question :)
As I was reading your reply, I was watching one of Leonard Susskind's videos and he happened to be explaining something involving the Lagrangian function and I think he also answered my question. I am going to go over that part of the video again. It's #4 in the Phys 25 (Classical Mechanics) series, where he discusses symmetry and the "action".