Chain rule for denominator in second order derivatives

[redtree]

TL;DR

What is the chain rule for switching the denominator of a second order derivative?

Given $\frac{d^2x}{dy^2}$, what is the chain rule for transforming to $\frac{d^2 x}{dz^2}$?

(This is not a homework question)

 

[FactChecker]

$dx/dz = dx/dy*dy/dz$ (chain rule)
So
$d^2x/d^2z = dx/dy*d^2y/d^2z + d^2x/dydz*dy/dz$ (product rule) (Corrected as @PeroK pointed out)

 

[PeroK]

$dx/dz = dx/dy*dy/dz$ (chain rule)
So
$d^2x/d^2z = dx/dy*d^2y/d^2z + d^2x/dxdz*dy/dz$ (product rule)

You've got a typo there. The second term should be $\frac{d^2x}{dydz}$.

 

[redtree]

Great. Thanks so much!