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

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Answer 1 · 2016-10-23T19:30:17

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

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

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

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