This chapter takes the study of calculus forward into more advanced topics involving multiple variable functions. In general, most functions that are found in the machine learning field are ones with many variables rather than just one. Quite often, we are trying to maximise some value or minimise some error function, so the ability to differentiate such functions and find their maxima and minima will be essential. This leads us to the methods of partial differentiation that enable us to find gradients in different planes as described in Sect. 6.1. We also briefly look at multiple integrals that will be needed when we look at probability distributions of multiple continuous random variables in Chap. 11 .