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