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.