Derivative of a function with respect to another function

[tahlaskerssen]

So I'm just having a bad night about Lagrangian and Hamiltonian mechanics and I was deriving them all over again and just wanted to try to differenciate the derivation with respect to velocity instead of x (just for fun)

But I kind of struggled with the math and I just want to get this right with the chain rule

If F is a function of x and V and V a function of x as well and I want to differentiate F with respect to V

dF/dV = dF/dx . dx/dV

Is it really this way?

I kind of took the formula from the post at the bottom here: -> https://www.physicsforums.com/showthread.php?t=282120

But i don't really understand what that comes from.. is ti true the equations that guy wrote? I've never seen the chain rule that way

 

[HallsofIvy]

I am not sure exactly what you are asking but it is certainly true that if U(x) and V(x) are functions of some common variable, x, then "the derivative of U with respect to V" is
\frac{dU}{dV}= \frac{\frac{dU}{dx}}{\frac{dV}{dx}}.