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So the general process is this:

y = arcsec(x)

sec(y) = x

dy/dx * sec(y)tan(y) = 1

dy/dx = 1/[sex(y)tan(y)]

sec(y) = x

And for tan(y) we use the Pythagorean identities:

tan^2(y) = sec^2(y) - 1

tan(y) = [sec^(y) - 1] ^(1/2)

So dy/dx = 1/[x(x^2-1)^(1/2)]

However, my calculus book has one minor difference in it's derivative, an absolute value:

dy/dx = 1/[abs(x)(x^2-1)^(1/2)]

Where does this absolute value come from?