Answer
$2$
Work Step by Step
Consider $\dfrac{\partial w}{\partial u}=\dfrac{\partial w}{\partial x}\dfrac{\partial x}{\partial u}+\dfrac{\partial w}{\partial y}\dfrac{\partial y}{\partial u}$
and $(y\cos xy+\sin y)(2u)+(x \cos xy+x\cos y)(v)=((uv)\cos (u^3v+uv^3)+\sin (uv))(2u)+((u^2+v^2) \cos (u^3v+uv^3)+(u^2+v^2)\cos (uv))(v)$
For the point, $u=0,v=1$:
Thus, $\dfrac{\partial w}{\partial v}=[(1)\cos (0+0)+\sin (0)](0)+((0+1) \cos (0)+(0+1)\cos (0))(1)$
or, $0+1+1=2$
