/ 1\ log|log(x) + -| \ x/
log(log(x) + 1/x)
Let .
The derivative of is .
Then, apply the chain rule. Multiply by :
Differentiate term by term:
The derivative of is .
Apply the power rule: goes to
The result is:
The result of the chain rule is:
Now simplify:
The answer is:
1 1 - - -- x 2 x ---------- 1 log(x) + - x
/ 2 \ | / 1\ | | |1 - -| | | 2 \ x/ | -|1 - - + ----------| | x 1 | | - + log(x)| \ x / ---------------------- 2 /1 \ x *|- + log(x)| \x /
3 / 1\ / 1\ / 2\ 2*|1 - -| 3*|1 - -|*|1 - -| 6 \ x/ \ x/ \ x/ 2 - - + ------------- + ----------------- x 2 1 /1 \ - + log(x) |- + log(x)| x \x / ----------------------------------------- 3 /1 \ x *|- + log(x)| \x /