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## Main Question or Discussion Point

Continue reading...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:

$$\frac{\partial f}{\partial x} = \frac{\partial f}{\partial u}\frac{\partial u}{\partial x} + \frac{\partial f}{\partial v}\frac{\partial v}{\partial x} \ \ \...