/ __________________\ | / sin(2*x) | log| / 1 - ------------ | \\/ 1 + sin(2*x) /
log(sqrt(1 - sin(2*x)/(1 + sin(2*x))))
Let .
The derivative of is .
Then, apply the chain rule. Multiply by :
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 a constant times a function is the constant times the derivative of the function.
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 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:
The result is:
Now plug in to the quotient rule:
So, the result is:
The result is:
The result of the chain rule is:
The result of the chain rule is:
Now simplify:
The answer is:
cos(2*x) cos(2*x)*sin(2*x) - ------------ + ----------------- 1 + sin(2*x) 2 (1 + sin(2*x)) ---------------------------------- sin(2*x) 1 - ------------ 1 + sin(2*x)
/ 2 2 2 \ | sin (2*x) 2*cos (2*x) 2*cos (2*x)*sin(2*x) | |- ------------ + ------------ - -------------------- + sin(2*x) | | 1 + sin(2*x) 1 + sin(2*x) 2 2 | | (1 + sin(2*x)) cos (2*x) | 2*|--------------------------------------------------------------- - ------------| | sin(2*x) 1 + sin(2*x)| | 1 - ------------ | \ 1 + sin(2*x) / ---------------------------------------------------------------------------------- 1 + sin(2*x)
/ 2 2 2 / 2 2 2 \\ | 7*sin(2*x) 6*cos (2*x) 6*sin (2*x) 6*cos (2*x)*sin(2*x) | sin (2*x) 2*cos (2*x) 2*cos (2*x)*sin(2*x) || |1 - ------------ - --------------- + --------------- + -------------------- 3*|- ------------ + ------------ - -------------------- + sin(2*x)|| | 1 + sin(2*x) 2 2 3 2 | 1 + sin(2*x) 1 + sin(2*x) 2 || | (1 + sin(2*x)) (1 + sin(2*x)) (1 + sin(2*x)) 2*cos (2*x) \ (1 + sin(2*x)) /| 4*|--------------------------------------------------------------------------- - --------------- + -------------------------------------------------------------------|*cos(2*x) | sin(2*x) 2 / sin(2*x) \ | | 1 - ------------ (1 + sin(2*x)) |1 - ------------|*(1 + sin(2*x)) | \ 1 + sin(2*x) \ 1 + sin(2*x)/ / -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1 + sin(2*x)