Apply the quotient rule, which is:
and .
To find :
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:
To find :
Apply the power rule: goes to
Now plug in to the quotient rule:
Now simplify:
The answer is:
2 -1 - cot (x) cot(x) ------------ - ------ x 2 x
/ 2 \ |1 + cot (x) cot(x) / 2 \ | 2*|----------- + ------ + \1 + cot (x)/*cot(x)| | x 2 | \ x / ----------------------------------------------- x
/ / 2 \ / 2 \ \ |/ 2 \ / 2 \ 3*cot(x) 3*\1 + cot (x)/ 3*\1 + cot (x)/*cot(x)| -2*|\1 + cot (x)/*\1 + 3*cot (x)/ + -------- + --------------- + ----------------------| | 3 2 x | \ x x / ---------------------------------------------------------------------------------------- x