sin(x) ---------- 1 - cos(x)
d / sin(x) \ --|----------| dx\1 - cos(x)/
Apply the quotient rule, which is:
and .
To find :
The derivative of sine is cosine:
To find :
Differentiate term by term:
The derivative of the constant is zero.
The derivative of a constant times a function is the constant times the derivative of the function.
The derivative of cosine is negative sine:
So, the result is:
The result is:
Now plug in to the quotient rule:
Now simplify:
The answer is:
2 cos(x) sin (x) ---------- - ------------- 1 - cos(x) 2 (1 - cos(x))
/ 2 \ | 2*sin (x) | | ----------- + cos(x) | | -1 + cos(x) 2*cos(x) | |1 - -------------------- - -----------|*sin(x) \ -1 + cos(x) -1 + cos(x)/ ----------------------------------------------- -1 + cos(x)
/ 2 \ 2 | 6*cos(x) 6*sin (x) | / 2 \ sin (x)*|-1 + ----------- + --------------| | 2*sin (x) | 2 | -1 + cos(x) 2| 3*|----------- + cos(x)|*cos(x) 3*sin (x) \ (-1 + cos(x)) / \-1 + cos(x) / ----------- - ------------------------------------------- - ------------------------------- + cos(x) -1 + cos(x) -1 + cos(x) -1 + cos(x) ---------------------------------------------------------------------------------------------------- -1 + cos(x)