_________ / 3/x\ / tan |-| \/ \2/
Let .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
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:
The result of the chain rule is:
Now simplify:
The answer is:
/ 2/x\\ _________ | 3*tan |-|| / 3/x\ |3 \2/| / tan |-| *|- + ---------| \/ \2/ \2 2 / ----------------------------- /x\ 2*tan|-| \2/
/ 2/x\\ _________ | 1 + tan |-|| / 3/x\ / 2/x\\ | \2/| 3* / tan |-| *|1 + tan |-||*|4 + -----------| \/ \2/ \ \2// | 2/x\ | | tan |-| | \ \2/ / ----------------------------------------------- 16
/ 2 \ | / 2/x\\ / 2/x\\| _________ | |1 + tan |-|| 20*|1 + tan |-||| / 3/x\ / 2/x\\ | /x\ \ \2// \ \2//| 3* / tan |-| *|1 + tan |-||*|16*tan|-| - -------------- + ----------------| \/ \2/ \ \2// | \2/ 3/x\ /x\ | | tan |-| tan|-| | \ \2/ \2/ / ----------------------------------------------------------------------------- 64