Answer
$$\dfrac{\partial P}{\partial x} =\dfrac{6}{\sqrt 5} \\ \dfrac{\partial P}{\partial y} =\dfrac{2}{\sqrt 5}$$
Work Step by Step
We have $\dfrac{\partial P}{\partial x} =\dfrac{ue^y}{\sqrt{u^2+v^2+w^2}}+\dfrac{vye^x}{\sqrt{u^2+v^2+w^2}}+\dfrac{wye^{xy}}{\sqrt{u^2+v^2+w^2}}$
For $x=0; y=2$:
$\dfrac{\partial P}{\partial x} =\dfrac{0}{\sqrt{(0)^2+(1)^2+(2)^2}}+\dfrac{(2)(2)(e^0)}{\sqrt{(0)^2+(1)^2+(2)^2}}+\dfrac{(1) (2) ( e^0)}{\sqrt{(0)^2+(1)^2+(2)^2}} \\=\dfrac{6}{\sqrt 5}$;
Now, $\dfrac{\partial P}{\partial y} =\dfrac{ux(e^y)}{\sqrt{u^2+v^2+w^2}}+\dfrac{v(e^x)}{\sqrt{u^2+v^2+w^2}}+\dfrac{wx(e^{xy})}{\sqrt{u^2+v^2+w^2}}$
For $x=0; y=2$:
$\dfrac{\partial P}{\partial y} =\dfrac{0}{\sqrt{(0)^2+(1)^2+(2)^2}}+\dfrac{(2) (e^0)}{\sqrt{(0)^2+(1)^2+(2)^2}}+\dfrac{0}{\sqrt{(0)^2+(1)^2+(2)^2}} \\=\dfrac{2}{\sqrt 5}$
