# f-x-y-z-y-2-z-3-1-x-xz-3-1-y-2-xy-2-1-z-3-find-f-x-f-y-f-z-

Question Number 303 by 123456 last updated on 25/Jan/15
$${f}\left({x},{y},{z}\right)=\frac{{y}^{\mathrm{2}} {z}^{\mathrm{3}} }{\mathrm{1}−{x}}+\frac{{xz}^{\mathrm{3}} }{\mathrm{1}−{y}^{\mathrm{2}} }+\frac{{xy}^{\mathrm{2}} }{\mathrm{1}−{z}^{\mathrm{3}} } \\$$$$\mathrm{find} \\$$$$\frac{\partial{f}}{\partial{x}}+\frac{\partial{f}}{\partial{y}}+\frac{\partial{f}}{\partial{z}} \\$$
Answered by prakash jain last updated on 20/Dec/14
$$\frac{\partial{f}}{\partial{x}}=−\frac{{y}^{\mathrm{2}} {z}^{\mathrm{3}} }{\left(\mathrm{1}−{x}\right)^{\mathrm{2}} }+\frac{{z}^{\mathrm{3}} }{\left(\mathrm{1}−{y}^{\mathrm{2}} \right)}+\frac{{y}^{\mathrm{2}} }{\mathrm{1}−{z}^{\mathrm{3}} } \\$$$$\frac{\partial{f}}{\partial{y}}=\frac{\mathrm{2}{yz}^{\mathrm{3}} }{\mathrm{1}−{x}}−\frac{\mathrm{2}{yz}^{\mathrm{3}} }{\left(\mathrm{1}−{y}^{\mathrm{2}} \right)^{\mathrm{2}} }+\frac{\mathrm{2}{xy}}{\mathrm{1}−{z}^{\mathrm{3}} } \\$$$$\frac{\partial{f}}{\partial{z}}=\frac{\mathrm{3}{y}^{\mathrm{2}} {z}^{\mathrm{2}} }{\mathrm{1}−{x}}+\frac{\mathrm{3}{xz}^{\mathrm{2}} }{\mathrm{1}−{z}^{\mathrm{2}} }−\frac{\mathrm{3}{z}^{\mathrm{2}} {y}^{\mathrm{3}} }{\left(\mathrm{1}−{z}^{\mathrm{3}} \right)^{\mathrm{2}} } \\$$$$\mathrm{Substituting}\:\mathrm{the}\:\mathrm{given}\:\mathrm{values}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expression} \\$$$$\mathrm{will}\:\mathrm{give}\:\mathrm{result}. \\$$