`f(x) = 2arcsin(x-1)` Find the derivative of the function

The derivative of function f with respect to x is denoted as` f'(x)` .

To take the derivative of the given function: `f(x) =2arcsin(x-1)` ,

we can apply the basic property: `d/(dx) [c*f(x)] = c * d/(dx) [f(x)]` .

  then` f'(x) = 2 d/(dx) (arcsin (x-1))`

  To solve for the `d/(dx) (arcsin(x-1))` , we consider the derivative formula of an inverse trigonometric function.

 For the derivative of inverse "sine" function, we follow:

`d/(dx) (arcsin (u)) = ((du)/(dx))/sqrt(1-u^2)`

To apply the formula with the given function, we let` u= x-1` then `(du)/(dx) = 1` .

`d/(dx) (arcsin(x-1))= 1/sqrt(1-(x-1)^2)`

 Then `f'(x)=2 d/(dx) (arcsin (x-1))` becomes:

`f'(x) =2 * 1/sqrt(1-(x-1)^2)`

`f'(x) =2 /sqrt(1-(x-1)^2)`

To further simplify, we can evaluate the exponent inside the radical:

`f'(x) =2/sqrt(1-(x^2-2x+1))`

Note: `(x-1)^2= (x-1)(x-1)`

Applying FOIL or distributive property:

`(x-1)(x-1)= x*x + x*(-1) + (-1)*x + (-1)(-1)`

                       `=x^2 –x –x+1`

                      ` =x^2 -2x+1`

Simplify the expression inside radical:

`f'(x) =2/sqrt(1-x^2+2x -1)`

`f'(x)=2/sqrt(-x^2+2x) or f'(x)=2/sqrt(2x-x^2)`