16.1 Partial and total derivatives
Let’s take a look at multivariable functions more closely! For the sake of simplicity, let f : ℝ2 →ℝ be our function of two variables. To emphasize the dependence on the individual variables, we often write
Here’s the trick: by fixing one of the variables, we obtain the two single-variable functions! That is, if x1 ∈ℝ2 is fixed, we have x→f(x1,x), and if x2 ∈ℝ2 is fixed, we have x→f(x,x2), both of which are well-defined univariate functions. Think about this as slicing the function graph with a plane parallel to the x−z or the y −z axes, as illustrated by Figure 16.1. The part cut out by the plane is a single-variable function.
We can define the derivative of these functions by the limit of difference quotients. These are called the partial derivatives:
(Keep in mind that x1 signifies...
