What is the derivative of (2+sinx) / (2x+cosx)?

Answer 1

Treat this as (2 + sin(x)) * (2x + cos(x))^-1 and use the product rule (derivative of first tem times the second term, plus the first term times the derivative of the second).

(2 + cos(x)) * (2x + cos(x))^-1 + (2 + sin(x)) *[ -1(2x + cos(x))^-2 (2 - sin(x)) ]

simplify from here
(2 + cos(x))(2x + cos(x))^-1 - (4 - sin^2(x)) (2x + cos(x))^-2
etc.

Answer 2

Quotient rule says:
[(cos x) (2x + cos x) - (2 + sin x) (2 - sin x)] / [(2x + cos x)^2]
= [2x cos x + cos^2 x - 4 + sin^2 x] / [(2x + cos x)^2]
= (2x cos x - 3) / (2x + cos x)^2.