If u = x²/(x² + y²), find ou/ox, du/dy.

# As per the guidelines we are entitled to solve one question at a time, please resubmit the other question if you wish to get it answered.

Given $u=\frac{x^2}{x^2+y^2}$

To find the value of $\frac{\partial u}{\partial x}$ and $\frac{\partial u}{\partial y}$ proceed as follows.

$\begin{array}{rcl}\frac{\partial u}{\partial x}& =& \frac{\partial \left({\displaystyle \frac{x^2}{x^2+y^2}}\right)}{\partial x}\\ & =& \frac{\left(x^2+y^2\right)\left(2x\right)-x^2\left(2x\right)}{{\left(x^2+y^2\right)}^2}\\ & =& \frac{\cancel{2x^3}+2x{y}^2-\cancel{2x^3}}{{\left(x^2+y^2\right)}^2}\\ & =& \frac{2x{y}^2}{{\left(x^2+y^2\right)}^2}\end{array}$

Therefore, the value of $\begin{array}{rcl}\frac{\partial u}{\partial x}& =& \frac{2x{y}^2}{{\left(x^2+y^2\right)}^2}\end{array}$