Question

Compute the second partial derivatives

Answer 1

$f(x,y)=\log(x-y)$

has first-order partial derivatives

$\dfrac{\partial f}{\partial x}=\dfrac1{x-y}$

$\dfrac{\partial f}{\partial y}=-\dfrac1{x-y}$

Then the second-order partial derivatives are

$\dfrac{\partial^2f}{\partial x^2}=-\dfrac1{(x-y)^2}$

$\dfrac{\partial^2f}{\partial x\partial y}=\dfrac1{(x-y)^2}$

$\dfrac{\partial^2f}{\partial y\partial x}=\dfrac1{(x-y)^2}$

$\dfrac{\partial^2f}{\partial y^2}=-\dfrac1{(x-y)^2}$