Calculus Question

Question

Prove this equation using the method of this example and a previous equation with x replaced by y.

(a) using the method of the example

Let y = tanh⁻¹ x. Then ____ = tanh y= (sinh y)/(cosh y)= (e^2y-1)/(e^2y+1)

Answer 1

exchanging x and y

have x=tanh(y) = [e^2y-1]/[e^2y+1]

e^2y(x-1)=-x-1

or y=tanh^-1x=(1/2)ln([x+1]/[1-x])

notice domain of tanh^-1x is -1<x so numerator is positive

and x<1 so denominator is positive