/ / 2 \\ |log\x - 1/| |-----------| \ 4 / ------------- 2 x + 1
(log(x^2 - 1)/4)/(x^2 + 1)
Apply the quotient rule, which is:
and .
To find :
Let .
The derivative of is .
Then, apply the chain rule. Multiply by :
Differentiate term by term:
Apply the power rule: goes to
The derivative of the constant is zero.
The result is:
The result of the chain rule is:
To find :
Differentiate term by term:
The derivative of the constant is zero.
The derivative of a constant times a function is the constant times the derivative of the function.
Apply the power rule: goes to
So, the result is:
The result is:
Now plug in to the quotient rule:
Now simplify:
The answer is:
/ 2 \ x x*log\x - 1/ ------------------- - ------------- / 2 \ / 2 \ 2 2*\x + 1/*\x - 1/ / 2 \ 2*\x + 1/
2 / 2 \ 2*x | 4*x | / 2\ -1 + ------- |-1 + ------|*log\-1 + x / 2 | 2| 2 -1 + x \ 1 + x / 2*x - ------------ + -------------------------- - ------------------ / 2\ / 2\ / 2\ / 2\ 2*\-1 + x / 2*\1 + x / \1 + x /*\-1 + x / ---------------------------------------------------------------- 2 1 + x
/ 2 / 2 \ / 2 \ / 2 \ \ | 4*x | 2*x | / 2\ | 2*x | | 4*x | | |-3 + ------- 6*|-1 + ------|*log\-1 + x / 3*|-1 + -------| 3*|-1 + ------| | | 2 | 2| | 2| | 2| | | -1 + x \ 1 + x / \ -1 + x / \ 1 + x / | x*|------------ - ---------------------------- + ------------------ + ------------------| | 2 2 / 2\ / 2\ / 2\ / 2\| | / 2\ / 2\ \1 + x /*\-1 + x / \1 + x /*\-1 + x /| \ \-1 + x / \1 + x / / ----------------------------------------------------------------------------------------- 2 1 + x