Find the derivative of the following functions.
(i) `((x + 1)/(x + 2))`
(ii) `(x^(2) + 1)/("sin" x)`
(iii) `(e^(x))/(1 + cos x)`

Explanation

(i) Let `f(x) = (x + 1)/(x + 2)`
`:. (d)/(dx) (f(x)) = (d)/(dx) ((x + 1)/(x + 2))`
`= ((x + 2) (d)/(dx) (x + 1) - (x + 1) (d)/(dx) (x + 2))/((x + 2)^(2))`
`= ((x + 2) (1) - (x + 1)(1))/((x + 2)^(2)) = (x + 2 - x - 1)/((x + 2)^(2)) = (1)/((x + 2)^(2))`
(ii) `f(x) (x^(2) + 1)/(sin x)`
`:. (d)/(dx) (f(x)) = (d)/(dx) ((x^(2) + 1)/(sin x)) = ("sin" (d)/(dx) (x^(2) + 1) - (x^(2) + 1) (d)/(dx) (sin x))/(sin^(2) x)`
`= ((sin x) 2x (x^(2) + 1) cos x)/(sin^(2) x) = (2x sin x - (x^(2) + 1) cos x)/(sin^(2) x)`
(iii) let `f(x) = (e^(x))/(1 + cos x)`
`:. (d)/(dx) (f(x)) = (d)/(dx) ((e^(x))/(1 + cos x)) = ((1 + cos x)(d)/(dx) (e^(x)) (d)/(dx) (1 + cos x))/((1 + cos x)^(2))`
`= ((1 + cos x) e^(x) - e^(x) (- sin x))/((1 + cos x)^(2))`
`= ((1 + cos x)e^(x) + e^(x) sin x)/((1 + cos x)^(2)) = (e^(x) (1 + cos x + sin x))/((1 + cos x)^(2))`