Chain rule for second derivatives

[ehrenfest]

Homework Statement

I cannot believe I do not know this!

$$dx/dt = dx/dt' dt'/dt$$ is the chain rule for first derivatives

$$d^2x/dt^2 = ?$$ is the chain rule for second derivatives

if it is complicated could you link me to a source that explains it please

Homework Equations

The Attempt at a Solution

 

[Hurkyl]

What's the problem with differentiating the first derivative?

 

[ehrenfest]

$$d^2x/dt^2 = d^2x/(dt'dt) dt'/dt + dx/dt d^2t'/dt^2$$?

The product rule was used. That mixed partial seems out of place. Is the above correct?

 

[HallsofIvy]

What makes you think there is a "chain rule for second derivatives"?

 

[ehrenfest]

What makes you think there is a "chain rule for second derivatives"?

Well, is there something wrong with what I posted? Maybe I just discovered it.

 

[Hurkyl]

$$d^2x/dt^2 = d^2x/(dt'dt) dt'/dt + dx/dt d^2t'/dt^2$$?

The product rule was used. That mixed partial seems out of place. Is the above correct?

Looks right. If you don't want the mixed partial, then why not apply the chain rule? After all, dx/dt' is merely a function... (Call it y, if you're having trouble thinking about it)