Week 15: Functions of several variables, Partial derivatives and their applications to optimization.
A function of variables, also called a function of several variables, with domain is a relation that assigns to every ordered tuple in a unique real number in . We denote this by each of the following types of notation. The range of is the set of all outputs of . It is a subset of , not . Functions of two variables have level curves, which are shown as curves in the xy−plane. However, when the function has three variables, the curves become surfaces, so we can define level surfaces for functions of three variables. ... Find the level surface for the function f(x,y,z)=4x2+9y2−z2 corresponding to c=1.
We'll assume you are familiar with the ordinary derivative \dfrac{df}{dx}dxdfstart fraction, d, f, divided by, d, x, end fraction from single variable calculus. I actually quite like this notation for the derivative, because you can interpret it as follows:

Interpret dxdxd, x as "a very tiny change in xxx".

Interpret dfdfd, f as "a very tiny change in the output of fff", where it is understood that this tiny change is whatever results from the tiny change dxdxd, x to the input.