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In this article, we will look closer at the concept of Hesse Matrices. To do so, we will need to understand how we can find the second-degree derivatives for multivariable functions.
Finding Partial Derivatives
We can start by remembering what is meant by taking the partial derivative. This means to find the derivatives with respect to one of the variables, with the others held constant. Let us consider when we have a multivariable function with two variables: x and y. We can take an illustrative example.
Imagine that we have the following function:
We can then start by finding the partial derivative regarding x:
We can see that we simply act like ‘y’ is a constant, like any other number, when we differentiate the function. We can then see how it would have looked if we took the partial derivative regarding y:
Finding the Second Degree Derivatives
We have now seen how we could find the partial derivatives. We can then try to see how it is possible to find the second-degree derivatives. Since it is with a multivariable function, there are more options. Let us consider the case, where we have a function with two variables, f(x,y). We know that we have the following derivatives:
Then we will have the following second-degree derivatives:
One important thing to know is that we have the following fact:
If we, on the other hand, had a function in three variables: f(x,y,z). Then we would have the following first derivatives:
Example of Second-Degree Derivatives
Let us try to take an example of where we find the second-degree derivatives of a multivariable function. Imagine that we have the following function:
We can start by finding the first partial derivatives:
We can then find the second-degree derivatives:
Defining the Hesse Matrix
We can now define the Hesse Matrix. It is similar to the gradient, where we gathered all the partial derivatives in a vector. The Hesse Matrix is simply a matrix consisting of all the second-degree derivatives. For a function in two variables, we have that:
