Question

calculate the derivative of

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

Answer 1

Let's note f the function defined for all the real values of x by
f(x)=(e^x-e^-x)/(e^x+e^-x)
For any real value of x e^x>0
e^-x>0
e^x+e^-x>0
e^x+e^-x is different from 0.
f is defined for any real value of x.
f is derivable for any real value of x as a quotient of two functions that are derivable for any real value of x.

using the quotient rule
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

multiplying and simplifying

=4/(e^x+e^-x)²

Answer 2

thought process behind derivative of e^ -x being e^ -x ? (
i understand derivative of e^x is e^x.)

Answer 3

thank you!