/x + 1\ log|-----| \x - 1/
log((x + 1)/(x - 1))
Let .
The derivative of is .
Then, apply the chain rule. Multiply by :
Apply the quotient rule, which is:
and .
To find :
Differentiate term by term:
The derivative of the constant is zero.
Apply the power rule: goes to
The result is:
To find :
Differentiate term by term:
The derivative of the constant is zero.
Apply the power rule: goes to
The result is:
Now plug in to the quotient rule:
The result of the chain rule is:
Now simplify:
The answer is:
/ 1 x + 1 \ (x - 1)*|----- - --------| |x - 1 2| \ (x - 1) / -------------------------- x + 1
/ 1 + x \ / 1 1 \ |1 - ------|*|- ----- - ------| \ -1 + x/ \ 1 + x -1 + x/ ------------------------------- 1 + x
/ 1 + x \ / 1 1 1 \ 2*|1 - ------|*|-------- + --------- + ----------------| \ -1 + x/ | 2 2 (1 + x)*(-1 + x)| \(1 + x) (-1 + x) / -------------------------------------------------------- 1 + x