I'll assume
@blamocur is right and you meant that [math]\frac{dw}{dt} = \frac{\partial w}{\partial x}(-\sin t) + \frac{\partial w}{\partial y}(\cos t)[/math]
Do you realize that f(x,y) is
unrelated to x(t) and y(t), so that [imath]\nabla w[/imath] doesn't depend on the latter? There is
no reason "why x is the j component of the vector", or "why ... the i component [is] y". They are both arbitrary.
The way w depends on x and y defines some
surface; the ways x and y depend on t define some
path on that surface. The derivative of x
with respect to t is not related to the derivative of w
with respect to x.
I can see why you might imagine a connection, since part of it does look related, but that is accidental.
Now, you need to replace x and y with their values in terms of t, in order to find [imath]\frac{dw}{dt}[/imath] as a function of t and answer the question.