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 :
Let .
The derivative of sine is cosine:
Then, apply the chain rule. Multiply by :
Apply the power rule: goes to
The result of the chain rule is:
To find :
Let .
The derivative of cosine is negative sine:
Then, apply the chain rule. Multiply by :
Apply the power rule: goes to
The result of the chain rule is:
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 :
Let .
The derivative of cosine is negative sine:
Then, apply the chain rule. Multiply by :
Apply the power rule: goes to
The result of the chain rule is:
To find :
Let .
The derivative of sine is cosine:
Then, apply the chain rule. Multiply by :
Apply the power rule: goes to
The result of the chain rule is:
Now plug in to the quotient rule:
Now simplify:
The answer is:
/ 2/ 2\\ 2*x*\-1 - cot \x //
/ 2/ 2\ 2 / 2/ 2\\ / 2\\ 2*\-1 - cot \x / + 4*x *\1 + cot \x //*cot\x //
/ 2/ 2\\ / / 2\ 2 2/ 2\ 2 / 2/ 2\\\ 8*x*\1 + cot \x //*\3*cot\x / - 4*x *cot \x / - 2*x *\1 + cot \x ///