$$if \quad u \quad = \left(\frac{\left(\arctan \left(xy\right)\right)}{\left(\sqrt{1+x^2+y^2}\right)}\right)$$
prove $$\frac{\partial{u}}{\partial{x}}=\frac{y}{(1+x^2)(\sqrt{1+x^2+y^2}}$$

Question

$$if \quad u \quad = \left(\frac{\left(\arctan \left(xy\right)\right)}{\left(\sqrt{1+x^2+y^2}\right)}\right)$$
prove $$\frac{\partial{u}}{\partial{x}}=\frac{y}{(1+x^2)(\sqrt{1+x^2+y^2}}$$

kc_kennylau · Answer

Firstly we have $$\frac{\partial}{\partial x}\arctan(xy)=\frac{y}{1+(xy)^2}$$

anonymous · Answer

$$\frac{\frac{y}{y^2x^2+1}\sqrt{1+x^2+y^2}-\frac{x}{\sqrt{1+x^2+y^2}}\arctan \left(xy\right)}{\left(\sqrt{1+x^2+y^2}\right)^2}$$ by applying quotient rule

kc_kennylau · Answer

Also we have $$\frac{\partial}{\partial x}\sqrt{1+x^2+y^2}=\frac{x}{\sqrt{1+x^2+y^2}}$$

kc_kennylau · Answer

ok how on earth would you prove that identity lol

alekos · Answer

I can't see a way through with this one.   Are you sure that is the correct expression for $$\delta u/\delta x$$