1 ------ /x\ tan|-| \2/
1/tan(x/2)
Let .
Apply the power rule: goes to
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 ------------- 2/x\ tan |-| \2/
/ 2/x\\ | 1 + tan |-|| / 2/x\\ | \2/| |1 + tan |-||*|-1 + -----------| \ \2// | 2/x\ | | tan |-| | \ \2/ / -------------------------------- /x\ 2*tan|-| \2/
3 2 / 2/x\\ / 2/x\\ 3*|1 + tan |-|| 5*|1 + tan |-|| 2/x\ \ \2// \ \2// -2 - 2*tan |-| - ---------------- + ---------------- \2/ 4/x\ 2/x\ tan |-| tan |-| \2/ \2/ ---------------------------------------------------- 4