Differentiate `(x)/(sinx)` w.r.t . `sinx.`

Answer

`(tan x-x)/(sin^(2)x)`

Explanation

`"Let "u=(x)/(sin x ) and v = sin x`
`therefore" "(du)/(dx)=(sin x.(d)/(dx)x -x. (d)/(dx) sin x)/((sin x)^(2))`
`=(sin x - x cos x)/(sin^(2)x)" ....(i)"`
`"And "(dv)/(dx)=(d)/(dx)( sin x )= cos x" ...(ii)"`
`therefore" "(du)/(dv)=(du//dx)/(dv//dx)=((sin x- x cos x)/sin^(2)x)/(cos x )`
`=(tan x-x)/(sin^(2)-x)`