sin(x) ------------- 2 (1 + cos(x))
d / sin(x) \ --|-------------| dx| 2| \(1 + cos(x)) /
Apply the quotient rule, which is:
and .
To find :
The derivative of sine is cosine:
To find :
Let .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
Differentiate term by term:
The derivative of the constant is zero.
The derivative of cosine is negative sine:
The result is:
The result of the chain rule is:
Now plug in to the quotient rule:
Now simplify:
The answer is:
2 cos(x) 2*sin (x) ------------- + ------------- 2 3 (1 + cos(x)) (1 + cos(x))
/ / 2 \ \ | |3*sin (x) | | | 2*|---------- + cos(x)| | | \1 + cos(x) / 4*cos(x) | |-1 + ----------------------- + ----------|*sin(x) \ 1 + cos(x) 1 + cos(x)/ -------------------------------------------------- 2 (1 + cos(x))
/ 2 \ 2 | 9*cos(x) 12*sin (x) | / 2 \ 2*sin (x)*|-1 + ---------- + -------------| |3*sin (x) | 2 | 1 + cos(x) 2| 6*|---------- + cos(x)|*cos(x) 6*sin (x) \ (1 + cos(x)) / \1 + cos(x) / -cos(x) - ---------- + ------------------------------------------- + ------------------------------ 1 + cos(x) 1 + cos(x) 1 + cos(x) --------------------------------------------------------------------------------------------------- 2 (1 + cos(x))