The language in this question is confusing!

I presume you need to find the derivative of:

y = tanh^-1 x + ln sqrt(1-x^2)

Since d/dx[tanh^-1 x] = -1/[(x - 1)(x + 1)]

Then

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

via the chain rule. (I'll let you simplify this.)

And I presume you need to find the derivative of:

y = x cosh^-1 x - sqrt(1-x^2)

Since d/dx[cosh^-1 x] = 1/sqrt[(x - 1)(x + 1)]

Then

dy/dx = 1 * cosh^-1x + x * 1/sqrt[(x - 1)(x + 1)] + (1/2) * 1/sqrt(1 - x^2) * (-2x)

via the product and chain rules. (Again, you need to simplify this.)

-Dan