Find the derivative of y when y= ln (arccosh x)

Question

I want to know how to find the derivative of y when y= ln (arccosh x) I know arccosh x = 1/[x^2 -1]^(1/2) So 1/[(arccosh x)^[2] [x^2 -1]^(1/2)] But the right answer is 1/[(arccosh x)^[2] [x^2 -1]^(1/2)]

Why?

Please help

Thanks all

Answer 1

You have : y = ln[arccosh(x)]. Define z = arccosh(x), then y = ln[z]; so the derivative of y with respect to z is just (dy/dz) = 1 / z. Now, the derivative of z with respect to x is (dz/dx) = 1 / Sqrt[x^2 - 1]. Now, apply the chain rule (dy/dx) = (dy/dz) (dz/dx).