2 cos (x) ------- log(x)
/ 2 \ d |cos (x)| --|-------| dx\ log(x)/
Apply the quotient rule, which is:
and .
To find :
Let .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
The derivative of cosine is negative sine:
The result of the chain rule is:
To find :
The derivative of is .
Now plug in to the quotient rule:
Now simplify:
The answer is:
2 cos (x) 2*cos(x)*sin(x) - --------- - --------------- 2 log(x) x*log (x)
2 / 2 \ cos (x)*|1 + ------| 2 2 \ log(x)/ 4*cos(x)*sin(x) - 2*cos (x) + 2*sin (x) + -------------------- + --------------- 2 x*log(x) x *log(x) ---------------------------------------------------------------- log(x)
/ 2 / 3 3 \ \ | cos (x)*|1 + ------ + -------| / 2 \ | | / 2 2 \ | log(x) 2 | 3*|1 + ------|*cos(x)*sin(x)| | 3*\sin (x) - cos (x)/ \ log (x)/ \ log(x)/ | 2*|4*cos(x)*sin(x) - --------------------- - ------------------------------ - ----------------------------| | x*log(x) 3 2 | \ x *log(x) x *log(x) / ----------------------------------------------------------------------------------------------------------- log(x)