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 \[ \begin{split} \frac{\partial^2 f}{\partial x^2}&=\frac{\partial }{\partial x}\frac{\partial f}{\partial x}=\frac{2(y^2-x^2)}{(x^2+y^2)^2}\\ \frac{\partial^2 f}{\partial y^2}&=\frac{\partial }{\partial y}\frac{\partial f}{\partial y}=\frac{2(x^2-y^2)}{(x^2+y^2)^2}\\ \frac{\partial^2 f}{\partial x\partial y}&=\frac{\partial }{\partial x}\frac{\partial f}{\partial y}=-\frac{4xy}{(x^2+y^2)^2}\\ \end{split} \]