Question

evaluate using product quotient chain sec(2theta)tan^2(theta)

Answer 1

This is tough.
So this combines both product and chain rules.

If you have a product of functions f*g,
then the derivative of that product is f’ * g + g’ * f.
So if f = sec(2x) and g = tan^2 (x),

then f’ = [2] [sec (2x) * tan (2x)]
(where the 2 comes from the chain rule)

and g’ = 2 * tan x [again chain rule].

So f’ * g + g’ * f =
([2] [sec (2x) * tan (2x)])* (tan^2 (x)) + [2 * tan x] * [sec(2x)].