cot(x) ------ 2 x + 1
cot(x)/(x^2 + 1)
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 :
Differentiate term by term:
The derivative of the constant is zero.
Apply the power rule: goes to
The result is:
Now plug in to the quotient rule:
Now simplify:
The answer is:
2 -1 - cot (x) 2*x*cot(x) ------------ - ---------- 2 2 x + 1 / 2 \ \x + 1/
/ / 2 \ \ | | 4*x | | | |-1 + ------|*cot(x) | | | 2| / 2 \| |/ 2 \ \ 1 + x / 2*x*\1 + cot (x)/| 2*|\1 + cot (x)/*cot(x) + -------------------- + -----------------| | 2 2 | \ 1 + x 1 + x / ------------------------------------------------------------------- 2 1 + x
/ / 2 \ / 2 \ \ | / 2 \ | 4*x | | 2*x | | | 3*\1 + cot (x)/*|-1 + ------| 12*x*|-1 + ------|*cot(x)| | | 2| / 2 \ | 2| | |/ 2 \ / 2 \ \ 1 + x / 6*x*\1 + cot (x)/*cot(x) \ 1 + x / | -2*|\1 + cot (x)/*\1 + 3*cot (x)/ + ----------------------------- + ------------------------ + -------------------------| | 2 2 2 | | 1 + x 1 + x / 2\ | \ \1 + x / / ------------------------------------------------------------------------------------------------------------------------- 2 1 + x