Question

FInd all second order partial derivatives: f(u,v) = ln(u^2 + v^2)

Answer 1

The second order partial derivatives of f(u,v) = ln(u^2 + v^2) are as follows:

fuu(u,v) = d^2/du^2 (ln(u^2 + v^2)) = -2u / (u^2 + v^2)

fuv(u,v) = d^2/du/dv (ln(u^2 + v^2)) = 0

fvv(u,v) = d^2/dv^2 (ln(u^2 + v^2)) = -2v / (u^2 + v^2)

So, the second partial derivatives of f(u,v) = ln(u^2 + v^2) are fuu(u,v) = -2u / (u^2 + v^2), fuv(u,v) = 0, and fvv(u,v) = -2v / (u^2 + v^2).