# Why do students mix up addition and multiplication, in the Multivariable Chain Rule with all partial derivatives?

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I have taught multivariate calculus for years. Annually, some student always swaps + and ×. They trump up

$\color{red}{\dfrac{\partial z}{\partial s} = (\dfrac{\partial z}{\partial x} + \dfrac{\partial x}{\partial s}) [\dfrac{\partial z}{\partial y} + \dfrac{\partial y}{\partial s}]}$

instead of this definition in James Stewart, Daniel Clegg, Saleem Watson. *Calculus Early Transcendentals* (2021 9 edn). Page 987.

This definition refers to “Theorem 1” scanned below.

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