Question

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

Answer 1

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

Answer 2

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

Answer 3

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

Answer 4

ok how on earth would you prove that identity lol

Answer 5

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