# Clairaut's Theorem on Higher Order Partial Derivatives

Recall from the Clairaut's Theorem on Higher Order Partial Derivatives page that Clairaut's theorem says that if $z = f(x, y)$ is a two variable real-valued function defined on a disk $\mathcal D$ containing the point $(a, b)$ then if the second order mixed partial derivatives of $f$ are continuous on $\mathcal D$ then:

(1)We will now look at some examples regarding Clairaut's theorem.

## Example 1

**Consider the function $z = x^2 y^4 - 2xy^4 + 2(xy + 1)$.** Verify that the second order mixed partial derivatives of this function are equal.**

We first compute the first partial derivatives of this function to get that:

(2)Now we compute the second order partial derivatives of this function:

(3)Therefore we see that $\frac{\partial^2 z}{\partial y \partial x} = \frac{\partial^2 z}{\partial x \partial y}$. This should not be surprising since the given function is merely a polynomial and polynomials are continuous everywhere.