Abstract
Derivatives are seen as the โeasyโ part of learning calculus: a few simple rules, and every function's derivatives are at your fingertips! But these basic techniques can turn bewildering if you are faced with much more complicated functions like a matrix determinant (what is a derivative โwith respect to a matrixโ anyway?), the solution of a differential equation, or a huge engineering calculation like a fluid simulation or a neural-network model. And needing such derivatives is increasingly common thanks to the growing prevalence of machine learning, large-scale optimization, and many other problems demanding sensitivity analysis of complex calculations. Although many techniques for generalizing and applying derivatives are known, that knowledge is currently scattered across a diverse literature, and requires students to put aside their memorized rules and re-learn what a derivative really is: linearization. In 2022 and 2023, Alan and I put together a one-month, 16-hour โMatrix Calculusโ course at MIT that refocuses differential calculus on the linear algebra at its heart, and we hope to remind you that derivatives are not a subject that is โdoneโ after your second semester of calculus