Question

partial differentiation of log(x^2 + y^2) is?

Answer 1

if you mean \(f(x,y) = \ln(x^2 + y^2)\) then \(f_x = \dfrac{1}{x^2 + y^2} . 2x\) by the chain rule; and ditto for \(f_y\).

you could test this by going \(e^{f(x,y)} = x^2 + y^2\), thence \(e^{f(x,y)} \, f_x = 2x \) such that \(f_x = \dfrac{2x}{x^2 + y^2} \)

if these are not natural logs, that second approach shows you how to factor that in.