Question

What the derivative of -csc^-1(2x+1/7) can any body help me to solve it?

Answer 1

`(dy)/(dx)=(-2)/(|2x+1/7|sqrt((2x+1/7)^2-1))`

Explanation:

We know that,

`(I)color(red)(d/(dX)(csc^-1X)=(-1)/(|X|sqrt(X^2-1))`

Here,

`y=csc^-1(2x+1/7)`

Let,

`y=csc^-1u , where, u=2x+1/7`

`=>(dy)/(du)=(-1)/(|u|sqrt(u^2-1)) and (du)/(dx)=2`

`"Using "color(blue)"Chain Rule :"`

`color(blue)((dy)/(dx)=(dy)/(du)*(du)/(dx)`

`(dy)/(dx)=(-1)/(|u|sqrt(u^2-1)) xx2`

`(dy)/(dx)=(-2)/(|u|sqrt(u^2-1))`

Subst. back ,

`(dy)/(dx)=(-2)/(|2x+1/7|sqrt((2x+1/7)^2-1))`

If, `y=-csc^-1(2x+1/7) ,then,`

`(dy)/(dx)=(2)/(|2x+1/7|sqrt((2x+1/7)^2-1))`