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