sin(6*x) ------------ 1 + cos(6*x)
sin(6*x)/(1 + cos(6*x))
Apply the quotient rule, which is:
and .
To find :
Let .
The derivative of sine is cosine:
Then, apply the chain rule. Multiply by :
The derivative of a constant times a function is the constant times the derivative of the function.
Apply the power rule: goes to
So, the result is:
The result of the chain rule is:
To find :
Differentiate term by term:
The derivative of the constant is zero.
Let .
The derivative of cosine is negative sine:
Then, apply the chain rule. Multiply by :
The derivative of a constant times a function is the constant times the derivative of the function.
Apply the power rule: goes to
So, the result is:
The result of the chain rule is:
The result is:
Now plug in to the quotient rule:
Now simplify:
The answer is:
2 6*cos(6*x) 6*sin (6*x) ------------ + --------------- 1 + cos(6*x) 2 (1 + cos(6*x))
/ 2 \ | 2*sin (6*x) | | ------------ + cos(6*x) | | 1 + cos(6*x) 2*cos(6*x) | 36*|-1 + ----------------------- + ------------|*sin(6*x) \ 1 + cos(6*x) 1 + cos(6*x)/ --------------------------------------------------------- 1 + cos(6*x)
/ / 2 \ \ | 2 | 6*cos(6*x) 6*sin (6*x) | / 2 \ | | sin (6*x)*|-1 + ------------ + ---------------| |2*sin (6*x) | | | 2 | 1 + cos(6*x) 2| 3*|------------ + cos(6*x)|*cos(6*x)| | 3*sin (6*x) \ (1 + cos(6*x)) / \1 + cos(6*x) / | 216*|-cos(6*x) - ------------ + ----------------------------------------------- + ------------------------------------| \ 1 + cos(6*x) 1 + cos(6*x) 1 + cos(6*x) / ----------------------------------------------------------------------------------------------------------------------- 1 + cos(6*x)