==פונקציה שהנגזרות החלקיות לא מתחלפות==
<math>f(x,y)=\begin{cases}xy\frac{x^2-y^2}{x^2+y^2} \ \text{if}\ x^2+y^2\neq 0 \\ 0 \ \text{else} \end{cases}</math>
<math>\frac{\partial}{\partial x} (\frac{\partial f}{\partial y}) (0,0) = \lim_{x\to 0} \frac{\frac{\partial f}{\partial y} (x,0)-\frac{\partial f}{\partial y} (0,0)}{x}</math>
<math>\frac{\partial f}{\partial y} (0,0)=0 , \ \ \frac{\partial f}{\partial y} (x,0)=\lim_{y\ to 0} \frac{f(x,y)-f(x,0)}{y}=x</math>
אז
<math>\frac{\partial}{\partial x} (\frac{\partial f}{\partial y}) (0,0) = \lim_{x\to 0} \frac{x-0}{x}=1</math>
אבל <math>f(x,y)=-f(y,x)</math> אז <math>\frac{\partial}{\partial y} (\frac{\partial f}{\partial x}) (0,0) = \lim_{y\to 0} \frac{-y-0}{y}=-1</math>
כלומר
<math>\frac{\partial}{\partial y} (\frac{\partial f}{\partial x}) (0,0) \neq \frac{\partial}{\partial x} (\frac{\partial f}{\partial y}) (0,0)</math>