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))##
