Question

Please help. Applying the chain rule, how do I find the derivative of f(x)=In(e^x-e^-x) (x>0) and then using this answer use the quotient rule to find the second derivative. I cannot do this so any answer greatly appreciated.

Answer 1

f(x) = ln(e^x - e^-x)

f'(x) = (e^x + e^-x)/(e^x - e^-x)

f''(x) = [(e^x - e^-x)(e^x - e^-x) - (e^x + e^-x)(e^x + e^-x)]/(e^x - e^-x)^2
= [(e^x - e^-x)^2 - (e^x + e^-x)^2]/(e^x - e^-x)^2