Homework Helper
Gold Member

Main Question or Discussion Point

Introduction
A frequent concern among students is how to carry out higher order partial derivatives where a change of variables and the chain rule are involved.  There is often uncertainty about exactly what the “rules” are.  This tutorial aims to clarify how the higher-order partial derivatives are formed in this case.
Note that in general second-order partial derivatives are more complicated than you might expect.  It’s important, therefore, to keep calm and pay attention to the details.
The General Case
Imagine we have a function $f(u, v)$ and we want to compute the partial derivatives with respect to $x$ and $y$ in terms of those with respect to $u$ and $v$.  Here we assume that $u, v$ may be expressed as functions of $x, y$.  The first derivative usually cause no problems.  We simply apply the chain rule:

etotheipi