Find the derivative of the following functions
(i) `x^(3) + cos x + sqrt(x)` (ii) `(sqrt(x) + 1)^(2)` (iii) `(x^(2) + 2x - 3)(x + 1)`

Explanation

Let `f(x) = x^(3) + cos x + sqrt(x)`
`:. (d)/(dx) (f(x)) = (d)/(dx) (x^(3) + cos x + sqrt(x))`
`= (d)/(dx) (x^(3)) + (d)/(dx) (cos x) + (d)/(dx) (sqrt(x)) [:' (d)/(dx) [f(x) + g (x) = (d)/(dx) (f(x)) + (d)/(dx) (g (x))]`
`3x^(3 -1) + (- sin x) + (1)/(2) x^(1//2 -1)`
`= 3x^(2) - sin x + (1)/(2 sqrt(x))`
(ii) Let `f(x) = (sqrt(X) + 1)^(2) = (sqrt(x))^(2) + (1)^(2) + 2sqrt(2) = x + 1+ 2 sqrt(x)`
`:. (d)/(dx) (f(x)) = (d)/(dx) (x + 1 + 2 sqrt(x))`
`= (d)/(dx) (x) + (d)/(dx) (1) + (d)/(dx) (2sqrt(x))`
`= 1 + 0 + (2)/(2 sqrt(x)) = 1 + (1)/(sqrt(x))`
(iii) Let `f(x) = (x^(2) + 2x - 3) (x + 1) = x^(3) + 2x^(2) - 3x + x^(2) + 2x - 3 = x^(3) - x - 3`
`:. (d)/(dx) (f(x)) = (d)/(dx) (x^(3) + 3x^(2) - x- 3) = (d)/(dx) (x^(3)) + (d)/(dx) (3x^(2)) - (d)/(dx) (x) - (d)/(dx) (3) = 3x^(2) 6x - 1`