Question

What is the derivative of this function `y=xcsc^-1x`?

Answer 1

Use the product rule and the derivative of `csc^-1x`

Explanation:

The derivative of `csc^-1x` depends on how you've defined it.

I use `y = csc^-1x` if and only if `cscy=x` and `- pi/2 < y < pi/2` with `y != 0`.

With this definition `d/dx(csc^-1 x ) = -1/(absx sqrt(x^2-1))`

For `f(x) = x csc^-1x`, we get

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

Alternatively

If you allow `0 < y < pi/2` or `pi < y < (3pi)/2`, then the absolute value is lost and we have

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

In this case

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