Partial Derivative of atan(xy/(1+x^2+y^2)^0.5)

[MrWarlock616]

Homework Statement

Prove that if $z=\arctan(\frac{xy}{\sqrt(1+x^2+y^2)})$ , then:

$\frac{\partial^2 z}{\partial x \partial y}=\frac{1}{(1+x^2+y^2)^\frac{3}{2}}$

Homework Equations

$\frac{d}{d x} (\arctan(x)) = \frac{1}{1+x^2}$

The Attempt at a Solution

Differentiating z partially w.r.t y, I got:

$\frac{\partial z}{\partial y} = \frac{x^3+x}{(1+x^2+y^2)^\frac{3}{2} (1+x^2+y^2+x^2y^2)}$

I'm pretty sure my working till here is correct. But differentiating again w.r.t x would be disastrous. Help me please. :)

 

[tiny-tim]

Hi MrWarlock616!

You can cancel a (1 + x²), top-and-bottom

 

[MrWarlock616]

EDIT: -nvm got it-