Find the derivative of the following functions w.r.t x
(i) ` sqrt(3x -2)` for `x gt 2/3`
(ii) ` (2x^(2) +3)^(5/3) (x+5)^(-1/3)`

Explanation

(i) Let f(x) = ` sqrt(3x -2)`
f(x) = ` 1/2 (3x -2)^(-1/2) d / (dx) (3x-2)`
` = 1/2 (3x-2) ^(-1/2) (3)`
` 3/(2sqrt(3x-2)) x lt 2/3`
(ii) ` y = (2x^(2) +3) ^(5/3) (x+5)^(-1/3)`
Differentiating w.r.t x , we get
`(dy)/(dx) = (2x^(2)+3)^(5/3) d/(dx) (x+5)^(-1/3) + (x+5)^(-1/3) d/(dx) ((2x^(2) +3)^(5/3))`
`= (2x^(2)+3)^(5/3)(-1/3)(x+5)^(-1/3-1) d/(dx) (x+5) + (x+5) + (x +5)^(-1/3) . 5/3(2x^(2)+3)^(5/3-1) d/(dx) (2x^(2)+3)`
`= (2x^(2)+3)^(5/3) (-1/3)(x+5)^(-4/3) 1+ (x=5)^(-1/3) .1 + (x+5)^(-1/3) 5/3 (2x^(2) +3)^(2/3)(4x)`
`1/3 (2x^(2)+3)^(2/3) (x+5)^(-1/3) (-2(2x^(2)+3)(x+5)^(-1) + 20x)`
` 1/3(2x^(2)+3)^(2/3)(x+5)^(-1/3) ( 20x - (2x^(2)+3)/(x+5))`
`= 1/3 ((2x^(2)+3)^(2/3)(18x^(2)+100x-3))/((x+5)^(4/3)), (x ne -5)`
(x+5)