How do you find the derivative of ##y=arcsin(1/x)##?

First, recall the identity ##d/dx[arcsinalpha] = 1/(sqrt(1 - x^2))##.

If this identity doesn't look familiar then I may recommend viewing a few videos from as they present a couple identities like this, and explain why they are true.

Differentiating ##arcsin (1/x)## is just a matter of using the identity above, as well as the :

##dy/dx = 1/sqrt(1-(1/x)^2) * d/dx[1/x]##

The derivative of ##1/x## is found using the power rule:

##dy/dx = 1/sqrt(1-1/x^2) * (-1/x^2)##

Now, all we need to do is simplify a bit:

##dy/dx = -1/(x^2sqrt((x^2-1)/x^2))##

##dy/dx = -1/(x^2/absxsqrt(x^2-1))##

##dy/dx=-1/(absxsqrt(x^2-1))##