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Second Partial Derivatives

Just like how we can take second derivatives of functions of one variable, we can also take second derivatives of functions of two variables. However, it is much more complicated to visualize and understand second derivatives in two dimensions.

Table of Contents​

Introduction​

We can take multiple partial derivatives of a function.

Consider the following function:

There are two partial derivatives, and for each of them, there are two more second partial derivatives. Recall that the notation for a second normal derivative, at least in Leibniz notation, is . For a second partial derivative:

  • If it's the same variable for both derivatives, we write .
  • If it's different variables (say then ), we write it as sort of an expansion of , which is .

Alternative Notations​

There are other notations for second partial derivatives. For example, we can write using Euler's notation as or . In my opinion, for higher-order derivatives, Euler's notation is more concise and easier to read.

We can show the different derivatives using a tree.

Let's first compute the first partial derivatives of . The things kept constant are shown in different colors to help understand the computation.

And the partial derivative with respect to :

Now, let's compute the second partial derivatives.

DerivativeComputation

Notice something interesting: . This is a property that certain functions have, called the Symmetry of Second Derivatives.

There's a formal definition for this property called Schwarz's Theorem, which states that if the second partial derivatives of a function are continuous in a region, then the mixed partial derivatives are equal. This theorem is discussed in the appendix.

Example Problem: Finding Higher-Order Derivatives​

The function is given by the following expression:

Find the following higher-order partial derivative:

(Source)

This problem is a bit more complicated than the previous one, but the process is the same.

First, make sure to do the partial derivatives in the correct order:

Now, let's compute the partial derivatives:

Thus, the fifth-order partial derivative is:

Summary and Next Steps​

In this section, we introduced the concept of second partial derivatives.

Here are the key points to remember:

  • Second partial derivatives are the derivatives of the partial derivatives of a function.
  • There are multiple notations for second partial derivatives, including Leibniz, Euler, and subscript notation.
  • The symmetry of second derivatives is a property that certain functions have, where the mixed partial derivatives are equal. This is formalized by Schwarz's Theorem.

In the next section, we will introduce the gradient of a function, which is a vector of partial derivatives.