This document demonstrates the mechanics, underlying a simple but not completely simplistic fully connected neural network (NN). In particular the multivariable mathematics involved in backpropogation is illustrated with a running example, which eschews the irritating handwavy approach found in many books, all of which seem to assume you can’t differentiate (some of these texts do cover differentiation but they assume the single variable case, while even simple NNs are multivariate in real life). Click here to view (Gave up trying to persuade Jekyll Markdown to display Mathematics).