/ 2 \ log\cot (x)/*x
log(cot(x)^2)*x
Apply the product rule:
; to find :
Let .
The derivative of is .
Then, apply the chain rule. Multiply by :
Let .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
There are multiple ways to do this derivative.
Rewrite the function to be differentiated:
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 :
The derivative of sine is cosine:
To find :
The derivative of cosine is negative sine:
Now plug in to the quotient rule:
The result of the chain rule is:
Rewrite the function to be differentiated:
Apply the quotient rule, which is:
and .
To find :
The derivative of cosine is negative sine:
To find :
The derivative of sine is cosine:
Now plug in to the quotient rule:
The result of the chain rule is:
The result of the chain rule is:
; to find :
Apply the power rule: goes to
The result is:
Now simplify:
The answer is:
/ 2 \ x*\-2 - 2*cot (x)/ / 2 \ ------------------ + log\cot (x)/ cot(x)
/ / 2\ \ | | / 2 \ | / 2 \| | | 2 \1 + cot (x)/ | 2*\1 + cot (x)/| 2*|x*|2 + 2*cot (x) - --------------| - ---------------| | | 2 | cot(x) | \ \ cot (x) / /
/ 2 / 2 \\ | / 2 \ | / 2 \ / 2 \|| | 2 3*\1 + cot (x)/ / 2 \ | \1 + cot (x)/ 2*\1 + cot (x)/|| 2*|6 + 6*cot (x) - ---------------- - 2*x*\1 + cot (x)/*|2*cot(x) + -------------- - ---------------|| | 2 | 3 cot(x) || \ cot (x) \ cot (x) //