Find the derivative of the following functions.
(i) `(x^(2) tan x)` (ii) `(x + cos x) (sqrt(x) + 1)` (iii) `x^(-3) (e^(x) + 5 "sin" x)`

Explanation

Let `f(x) = x^(2) tan x`
`:. (d)/(dx) (f(x)) = (d)/(dx) (x^(2) tan x)`
`tan x (d)/(dx) (x)^(2) + x^(2) (d)/(dx) (tan x)`
`= 2x tan x + sec^(2) x.x^(2)`
`= x (2 tan x + x sec^(2) x)`
(ii) Let `f(x) = (x + cos x) (sqrt(x) + 1) = x sqrt(x) + sqrt(x) cos x + x + cos x = x^(3//2) + sqrt(x) cos x + x + cos x`
`:. (d)/(dx) (f(x)) = (d)/(dx)) (x^(3//2) + sqrt(x) cos + x + cos x)`
`= (d)/(dx) (x^(3//2)) + (d)/(dx) (sqrt(x) cos x) + (d)/(dx) (x) + (d)/(dx) (cos x)`
`= (3)/(2) x^((3)/(2) 1) + sqrt(x) (d)/(dx) (cos x) + cos x (d)/(dx) (sqrt(x)) + 1 + (- sin x)` ,
`= (3)/(2) x^((1)/(2)) + sqrt(x) sqrt(x) (-sin x_ + cos x (1)/(2) x^(-1//2) + 1 - sin x`
`= (3)/(2) sqrt(X) - sqrt(x) sin x (cos x)/(2sqrt(x)) + 1 - sin x`
(iii) Let `f(x) = x^(-3) (e^(x) + 5 sin x) = x^(-3) e^(x)+ 5x^(-3) sin x`
`:. (d)/(dx) (f(x)) = (d)/(dx) (x^(-3) e^(x) + 5x^(-3) sin x`
`:. (d)/(dx) (f(x)) = (d)/(dx) (x^(-3) e^(x) + 5x^(-3) sin x)`
`= (d)/(dx) (x^(-3) e^(x)) + (d)/(dx) (5 x^(-3) sin x) = x^(-3) + e^(x) (d)/(dx) (x^(-3) (d)/(dx) (sin x) + 5 sin (d)/(dx) (x^(-3))`
`x^(-3) e^(x) - 3e^(-4) + 5x^(-3) cos x - 15 sin x.x^(-4) = x^(-3) (e^(x) + 5 cos x) - 3x^(-4) (e^(x) + 5 sin x)`