Question

Section 1, Part A, Session 4 worked example: why does sin2x derive to 2cos2x? I don't understand any process that would benefit from somehow converting sine to cosine.

Answer 1

How about the process of finding the tangent to the curve at a given point? Your not converting sine to cosine your finding the function to give you the slope of the tangent to the graph of sin(2x) at any point you choose.
It derives because of the chain rule `(F(g(x))) = F'(g(x))g'(x) in this case it's
F(x) = sin(x) and g(x) = 2x so F'(x) = cosx and g'(x) = 2
put them together you have 2cos(2x), so for example to find the tangent to the sin(2x) curve at say x=0 you would have the slope of the tangent equal to 2cos(2.0) = 2.