Beräkna alla andra ordningens partialderivator till funktionen
\[
f(x,y)=\ln(x^2+y^2)
\]
\[
\begin{split}
\frac{\partial^2 f}{\partial x^2}&=\frac{2(y^2-x^2)}{(x^2+y^2)^2},\\
\frac{\partial^2 f}{\partial y^2}&=\frac{2(x^2-y^2)}{(x^2+y^2)^2},\\
\frac{\partial^2 f}{\partial x\partial y}&=\frac{(-4xy)}{(x^2+y^2)^2}=\frac{\partial^2 f}{\partial y\partial x}
\end{split}
\]
Vi börjar med att beräkna första ordningens partialderivator:
\[
\frac{\partial f}{\partial x}=\frac{2x}{(x^2+y^2)},\quad\frac{\partial f}{\partial y}=\frac{2y}{(x^2+y^2)},\quad
\]
Sedan beräknar vi dessa derivators partialderivator och får
\[
\begin{split}
\frac{\partial^2 f}{\partial x^2}&=\frac{\partial }{\partial x}\frac{\partial f}{\partial x}=\frac{2(y^2-x^2)}{(x^2+y^2)^2}\\
\frac{\partial^2 f}{\partial y^2}&=\frac{\partial }{\partial y}\frac{\partial f}{\partial y}=\frac{2(x^2-y^2)}{(x^2+y^2)^2}\\
\frac{\partial^2 f}{\partial x\partial y}&=\frac{\partial }{\partial x}\frac{\partial f}{\partial y}=-\frac{4xy}{(x^2+y^2)^2}\\
\end{split}
\]