I understand that if a differentiable function \(\displaystyle z=f(g(x,y))\) and x, y are the independent variables, then

\(\displaystyle \frac{\partial{g}}{\partial{x}} = \lim_{\delta{x}\to{0}}\left(\frac{\delta{g}}{\delta{x}}\right)\) and \(\displaystyle \frac{\partial{f}}{\partial{g}} = \lim_{\delta{g}\to{0}}\left(\frac{\delta{f}}{\delta{g}}\right)\), so \(\displaystyle \frac{\partial{f}}{\partial{x}} = \frac{\partial{f}}{\partial{g}}.\frac{\partial{g}}{\partial{x}}\), keeping y constant

and

\(\displaystyle \frac{\partial{g}}{\partial{y}} = \lim_{\delta{y}\to{0}}\left(\frac{\delta{g}}{\delta{y}}\right)\) and \(\displaystyle \frac{\partial{f}}{\partial{g}} = \lim_{\delta{g}\to{0}}\left(\frac{\delta{f}}{\delta{g}}\right)\), so \(\displaystyle \frac{\partial{f}}{\partial{y}} = \frac{\partial{f}}{\partial{g}}.\frac{\partial{g}}{\partial{y}}\), keeping x constant

now, if I let \(\displaystyle u=g(x, y)\) can someone show/explain using the above method what \(\displaystyle \frac{\partial{z}}{\partial{u}}\) would be?.

Thanks.