# Derivative *of* a partial derivative

• Pollywoggy
In summary, the action is a vector that represents the change in the total energy of a system over time. It is always perpendicular to the direction of motion.f

## 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

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

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.

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".