Answer
$2$
Work Step by Step
Since, we have $\dfrac{\partial w}{\partial u}=\dfrac{\partial w}{\partial x}\dfrac{\partial x}{\partial u}+\dfrac{\partial w}{\partial y}\dfrac{\partial y}{\partial u}$
or, $=(y\cos xy+\sin y)(2u)+(x \cos xy+x\cos y)(v)$
or, $=((uv)\cos (u^3v+uv^3)+\sin (uv))(2u)+((u^2+v^2) \cos (u^3v+uv^3)+(u^2+v^2)\cos (uv))(v)$
Now, at point $u=0,v=1$, we get
$\dfrac{\partial w}{\partial v}=[(1)\cos (0+0)+\sin (0)](0)+((0+1) \cos (0)+(0+1)\cos (0))(1)=0+1+1=2$
