# Chain rule for second derivatives

1. Sep 13, 2007

### ehrenfest

1. The problem statement, all variables and given/known data
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

2. Relevant equations

3. The attempt at a solution

2. Sep 13, 2007

### Hurkyl

Staff Emeritus
What's the problem with differentiating the first derivative?

3. Sep 13, 2007

### 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?

4. Sep 14, 2007

### HallsofIvy

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

5. Sep 14, 2007

### ehrenfest

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

6. Sep 14, 2007

### Hurkyl

Staff Emeritus
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)