Find `(dy)/(dx)` for the following :
(i) `y=x^(7//2)` (ii) `y=x^(-3)`
`(iii) y=x` (iv) `y=x^(5)+x^(3)+4x^(1//2)+7`
(v) `y=5x^(4)+6x^(3//2)+9x` (vi) `y=ax^(2)+bx+C`
`(Vii) `y=3x^(5)-3x-1/x`

Answer

`2 tan^(-1) ((sigmaq_(0))/(2in_(0) mg))`

Explanation

From work energy theorem (WET)

`W_(EF)+W_(mg)=Delta KE`
`q_(0) xx sigma/(2 in_(0))xxL sin theta-mgL`
`(1- cos theta)=0`
`=q_(0)xx sigma/(2 in_(0))xxLxx2 "sin" theta/2 "cos" theta/2`
`=mgLxx2 "sin"^(2) (2 theta)/2, "tan" theta/2=sigma/(2 in_(0))xxq_(0)/(mg)`