helpppp! calculus

Question

If f(x) = (e ^ x + e ^ (- x))/2 with domain confined to the interval [0, 0), find the derivative of the inverse function by using the formula (f ^ - 1)' * (x) = 1/(f' * (f ^ - 1 * (x)))

Answer 1

If the domain is [0,0) then question is moot.

f(x)=(e^x+e^-x)/2=cosh(x)

has two branches of inverse function

set x=(e^y+e^-y)/2

2x=e^y+e^-y

2xe^y=e^2y+1

x²+2xe^y=e^2y+1+x² ,,,,adding x² to left and right side

x²-1=(e^y-x)²

e^y=x±√(x²-1)

y=cosh^-1(x)=ln(x±√(x²-1)) ,,,,two branches

note since -ln(x+√(x²-1))=ln(x+√(x²-1))^-1=ln(x-√(x²-1))

can rewrite

cosh^-1(x)=±ln(x+√(x²-1))

d(chosh(x))/dx=±[x+√(x²-1))^-1(1-x(x²-1)^(-1/2))]

d(cosh^-1(x))/dx=±(x²-1)^(-1/2)

and since also d(cosh^-1(x))/dx=1/sinh(cosh^-1(x))

sinh(cosh^-1(x))=±√(x²-1)