2/ /x\\ log |tan|-|| \ \6//
d / 2/ /x\\\ --|log |tan|-||| dx\ \ \6///
Let .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
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:
The result of the chain rule is:
Now simplify:
The answer is:
/ 2/x\\ | tan |-|| |1 \6/| / /x\\ 2*|- + -------|*log|tan|-|| \6 6 / \ \6// --------------------------- /x\ tan|-| \6/
/ 2/x\ / 2/x\\ / /x\\\ | 1 + tan |-| |1 + tan |-||*log|tan|-||| / 2/x\\ | / /x\\ \6/ \ \6// \ \6//| |1 + tan |-||*|2*log|tan|-|| + ----------- - -------------------------| \ \6// | \ \6// 2/x\ 2/x\ | | tan |-| tan |-| | \ \6/ \6/ / ----------------------------------------------------------------------- 18
/ 2 2 \ | / 2/x\\ / 2/x\\ / 2/x\\ / /x\\ / 2/x\\ / /x\\| | 3*|1 + tan |-|| 6*|1 + tan |-|| 4*|1 + tan |-||*log|tan|-|| 2*|1 + tan |-|| *log|tan|-||| / 2/x\\ | \ \6// / /x\\ /x\ \ \6// \ \6// \ \6// \ \6// \ \6//| |1 + tan |-||*|- ---------------- + 4*log|tan|-||*tan|-| + --------------- - --------------------------- + ----------------------------| \ \6// | 3/x\ \ \6// \6/ /x\ /x\ 3/x\ | | tan |-| tan|-| tan|-| tan |-| | \ \6/ \6/ \6/ \6/ / ---------------------------------------------------------------------------------------------------------------------------------------- 108