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Partial Differentiation and the Chain rule

The problem states: Find dw/dt (a) using the appropriate chain rule and (b) by converting w to a function of t before differentiating.

w = xy x = s sin t, y = cos t

the solution in my solution manual goes like this:

a) using the chain rule they come up with: 2y cos t + x(-sin t) = 2y cos t - x sin t =

2(cos^2t - sin^2t) = 2 cos 2t

b) w = 2 sin t cos t = sin 2t, dw/dt = 2 cos 2t

I cannot follow this solution at all. I'm not certain how to even begin to solve it.
Could someone help me make sense of this by showing me how they go from one step to the other?

Solution Preview

w=xy, x=2sint, y=cost
Let me explain (a).
by the product rule of differentiation, (xy)'=xy'+yx'. Now x=2sint, then x'=2cost because (sint)'=cost is a basic derivative formula. y=cost, then y'=-sint is also a basic ...

Solution Summary

Partial Differentiation and the Chain rule are investigated.