______________ / 1 - sin(2*x) / ------------ \/ 1 + sin(2*x)
/ ______________\ d | / 1 - sin(2*x) | --| / ------------ | dx\\/ 1 + sin(2*x) /
Let .
Apply the power rule: goes to
Then, apply the chain rule. Multiply by :
Apply the quotient rule, which is:
and .
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.
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:
So, the result is:
The result 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:
The result of the chain rule is:
Now simplify:
The answer is:
______________ / 1 - sin(2*x) / cos(2*x) (1 - sin(2*x))*cos(2*x)\ / ------------ *(1 + sin(2*x))*|- ------------ - -----------------------| \/ 1 + sin(2*x) | 1 + sin(2*x) 2 | \ (1 + sin(2*x)) / ---------------------------------------------------------------------------- 1 - sin(2*x)
/ 2 \ | / -1 + sin(2*x)\ 2 2 / -1 + sin(2*x)\ 2 / -1 + sin(2*x)\ | ___________________ | 2 |-1 + -------------| *cos (2*x) 2*cos (2*x)*|-1 + -------------| 2*cos (2*x)*|-1 + -------------| 2 | / -(-1 + sin(2*x)) | 4*cos (2*x) \ 1 + sin(2*x)/ \ 1 + sin(2*x)/ 2*(-1 + sin(2*x))*sin(2*x) \ 1 + sin(2*x)/ 4*cos (2*x)*(-1 + sin(2*x))| / ----------------- *|-2*sin(2*x) - ------------ + ------------------------------- - -------------------------------- + -------------------------- + -------------------------------- + ---------------------------| \/ 1 + sin(2*x) | 1 + sin(2*x) -1 + sin(2*x) 1 + sin(2*x) 1 + sin(2*x) -1 + sin(2*x) 2 | \ (1 + sin(2*x)) / ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -1 + sin(2*x)
/ / 2 2 \ / 2 2 \ / 2 2 \ \ | | 2*cos (2*x) (-1 + sin(2*x))*sin(2*x) 2*cos (2*x)*(-1 + sin(2*x))| | 2*cos (2*x) (-1 + sin(2*x))*sin(2*x) 2*cos (2*x)*(-1 + sin(2*x))| 3 2 / -1 + sin(2*x)\ | 2*cos (2*x) (-1 + sin(2*x))*sin(2*x) 2*cos (2*x)*(-1 + sin(2*x))| 2 | | 8*|-sin(2*x) - ------------ + ------------------------ + ---------------------------| 8*|-sin(2*x) - ------------ + ------------------------ + ---------------------------| / -1 + sin(2*x)\ 2 2 / -1 + sin(2*x)\ / -1 + sin(2*x)\ 2 6*|-1 + -------------|*|-sin(2*x) - ------------ + ------------------------ + ---------------------------| / -1 + sin(2*x)\ / -1 + sin(2*x)\ / -1 + sin(2*x)\ 2 2 / -1 + sin(2*x)\| ___________________ | | 1 + sin(2*x) 1 + sin(2*x) 2 | | 1 + sin(2*x) 1 + sin(2*x) 2 | 2 |-1 + -------------| *cos (2*x) 2 8*cos (2*x)*|-1 + -------------| 6*|-1 + -------------| *cos (2*x) \ 1 + sin(2*x)/ | 1 + sin(2*x) 1 + sin(2*x) 2 | 4*|-1 + -------------|*sin(2*x) 4*|-1 + -------------|*sin(2*x) 6*|-1 + -------------| *cos (2*x) 8*cos (2*x)*|-1 + -------------|| / -(-1 + sin(2*x)) | \ (1 + sin(2*x)) / 4*(-1 + sin(2*x)) \ (1 + sin(2*x)) / 24*sin(2*x) 24*cos (2*x) \ 1 + sin(2*x)/ 24*cos (2*x)*(-1 + sin(2*x)) 24*(-1 + sin(2*x))*sin(2*x) \ 1 + sin(2*x)/ \ 1 + sin(2*x)/ \ (1 + sin(2*x)) / \ 1 + sin(2*x)/ \ 1 + sin(2*x)/ \ 1 + sin(2*x)/ \ 1 + sin(2*x)/| / ----------------- *|-4 - ------------------------------------------------------------------------------------- + ----------------- + ------------------------------------------------------------------------------------- + ------------ + --------------- - ------------------------------- - ---------------------------- - --------------------------- - -------------------------------- - --------------------------------- - ---------------------------------------------------------------------------------------------------------- - ------------------------------- + ------------------------------- + --------------------------------- + --------------------------------|*cos(2*x) \/ 1 + sin(2*x) | -1 + sin(2*x) 1 + sin(2*x) 1 + sin(2*x) 1 + sin(2*x) 2 2 3 2 2 2 -1 + sin(2*x) -1 + sin(2*x) 1 + sin(2*x) (1 + sin(2*x))*(-1 + sin(2*x)) (1 + sin(2*x))*(-1 + sin(2*x)) | \ (1 + sin(2*x)) (-1 + sin(2*x)) (1 + sin(2*x)) (1 + sin(2*x)) (-1 + sin(2*x)) (-1 + sin(2*x)) / -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -1 + sin(2*x)
/ 2 2 2 \ | / 2 2 \ / 2 2 \ / 2 2 4 4 2 2 2 2 \ / 2 2 \ / 2 2 4 4 2 2 2 2 \ / 2 2 \ / 2 2 \ / 2 2 \ / 2 2 \ / 2 2 4 4 2 2 2 2 \ / 2 2 \ / 2 2 \ / 2 2 \ / 2 2 \ / 2 2 \ / 2 2 \ 2 / 2 2 \ / 2 2 \ / 2 2 \ 3 / 2 2 \ 2 / 2 2 \ / 2 2 \ / 2 2 \ / 2 2 \ / 2 2 \ / 2 2 \ 2 / 2 2 \ / 2 2 \ / 2 2 \| | | 2*cos (2*x) (-1 + sin(2*x))*sin(2*x) 2*cos (2*x)*(-1 + sin(2*x))| | 2*cos (2*x) (-1 + sin(2*x))*sin(2*x) 2*cos (2*x)*(-1 + sin(2*x))| | 8*cos (2*x) 6*sin (2*x) 24*cos (2*x) (-1 + sin(2*x))*sin(2*x) 24*cos (2*x)*(-1 + sin(2*x)) 6*sin (2*x)*(-1 + sin(2*x)) 8*cos (2*x)*(-1 + sin(2*x)) 36*cos (2*x)*sin(2*x) 36*cos (2*x)*(-1 + sin(2*x))*sin(2*x)| | 2*cos (2*x) (-1 + sin(2*x))*sin(2*x) 2*cos (2*x)*(-1 + sin(2*x))| | 8*cos (2*x) 6*sin (2*x) 24*cos (2*x) (-1 + sin(2*x))*sin(2*x) 24*cos (2*x)*(-1 + sin(2*x)) 6*sin (2*x)*(-1 + sin(2*x)) 8*cos (2*x)*(-1 + sin(2*x)) 36*cos (2*x)*sin(2*x) 36*cos (2*x)*(-1 + sin(2*x))*sin(2*x)| 5 2 2 | 2*cos (2*x) (-1 + sin(2*x))*sin(2*x) 2*cos (2*x)*(-1 + sin(2*x))| | 2*cos (2*x) (-1 + sin(2*x))*sin(2*x) 2*cos (2*x)*(-1 + sin(2*x))| 3 | -1 + sin(2*x) 6*sin(2*x) 6*cos (2*x) 6*cos (2*x)*(-1 + sin(2*x)) 6*(-1 + sin(2*x))*sin(2*x)| | 2*cos (2*x) (-1 + sin(2*x))*sin(2*x) 2*cos (2*x)*(-1 + sin(2*x))| / -1 + sin(2*x)\ 4 2 / -1 + sin(2*x)\ | 8*cos (2*x) 6*sin (2*x) 24*cos (2*x) (-1 + sin(2*x))*sin(2*x) 24*cos (2*x)*(-1 + sin(2*x)) 6*sin (2*x)*(-1 + sin(2*x)) 8*cos (2*x)*(-1 + sin(2*x)) 36*cos (2*x)*sin(2*x) 36*cos (2*x)*(-1 + sin(2*x))*sin(2*x)| | 2*cos (2*x) (-1 + sin(2*x))*sin(2*x) 2*cos (2*x)*(-1 + sin(2*x))| | -1 + sin(2*x) 6*sin(2*x) 6*cos (2*x) 6*cos (2*x)*(-1 + sin(2*x)) 6*(-1 + sin(2*x))*sin(2*x)| | -1 + sin(2*x) 6*sin(2*x) 6*cos (2*x) 6*cos (2*x)*(-1 + sin(2*x)) 6*(-1 + sin(2*x))*sin(2*x)| | 2*cos (2*x) (-1 + sin(2*x))*sin(2*x) 2*cos (2*x)*(-1 + sin(2*x))| 2 | -1 + sin(2*x) 6*sin(2*x) 6*cos (2*x) 6*cos (2*x)*(-1 + sin(2*x)) 6*(-1 + sin(2*x))*sin(2*x)| 2 / -1 + sin(2*x)\ | 2*cos (2*x) (-1 + sin(2*x))*sin(2*x) 2*cos (2*x)*(-1 + sin(2*x))| 2 / -1 + sin(2*x)\ 2 | 2*cos (2*x) (-1 + sin(2*x))*sin(2*x) 2*cos (2*x)*(-1 + sin(2*x))| / -1 + sin(2*x)\ | 2*cos (2*x) (-1 + sin(2*x))*sin(2*x) 2*cos (2*x)*(-1 + sin(2*x))| 2 | -1 + sin(2*x) 6*sin(2*x) 6*cos (2*x) 6*cos (2*x)*(-1 + sin(2*x)) 6*(-1 + sin(2*x))*sin(2*x)| 2 3 2 3 / -1 + sin(2*x)\ 2 | 2*cos (2*x) (-1 + sin(2*x))*sin(2*x) 2*cos (2*x)*(-1 + sin(2*x))| 4 / -1 + sin(2*x)\ 2 | -1 + sin(2*x) 6*sin(2*x) 6*cos (2*x) 6*cos (2*x)*(-1 + sin(2*x)) 6*(-1 + sin(2*x))*sin(2*x)| 3 2 / -1 + sin(2*x)\ | -1 + sin(2*x) 6*sin(2*x) 6*cos (2*x) 6*cos (2*x)*(-1 + sin(2*x)) 6*(-1 + sin(2*x))*sin(2*x)| | 2*cos (2*x) (-1 + sin(2*x))*sin(2*x) 2*cos (2*x)*(-1 + sin(2*x))| 2 | 2*cos (2*x) (-1 + sin(2*x))*sin(2*x) 2*cos (2*x)*(-1 + sin(2*x))| 2 2 / -1 + sin(2*x)\ | -1 + sin(2*x) 6*sin(2*x) 6*cos (2*x) 6*cos (2*x)*(-1 + sin(2*x)) 6*(-1 + sin(2*x))*sin(2*x)| 2 / -1 + sin(2*x)\ | 2*cos (2*x) (-1 + sin(2*x))*sin(2*x) 2*cos (2*x)*(-1 + sin(2*x))| 2 3 / -1 + sin(2*x)\ 2 | 2*cos (2*x) (-1 + sin(2*x))*sin(2*x) 2*cos (2*x)*(-1 + sin(2*x))| / -1 + sin(2*x)\ | 2*cos (2*x) (-1 + sin(2*x))*sin(2*x) 2*cos (2*x)*(-1 + sin(2*x))| 2 2 / -1 + sin(2*x)\ | 2*cos (2*x) (-1 + sin(2*x))*sin(2*x) 2*cos (2*x)*(-1 + sin(2*x))|| | 160*|-sin(2*x) - ------------ + ------------------------ + ---------------------------| 64*|-sin(2*x) - ------------ + ------------------------ + ---------------------------| 64*|-sin(2*x) - ------------ + ------------ + --------------- + ------------------------ - ---------------------------- - --------------------------- + --------------------------- + --------------------- - -------------------------------------| 64*|-sin(2*x) - ------------ + ------------------------ + ---------------------------| 64*|-sin(2*x) - ------------ + ------------ + --------------- + ------------------------ - ---------------------------- - --------------------------- + --------------------------- + --------------------- - -------------------------------------| / -1 + sin(2*x)\ 4 / -1 + sin(2*x)\ 4 4 / -1 + sin(2*x)\ 384*cos (2*x)*|-sin(2*x) - ------------ + ------------------------ + ---------------------------| 384*|-sin(2*x) - ------------ + ------------------------ + ---------------------------|*sin(2*x) / -1 + sin(2*x)\ 4 96*|-1 + ------------- + ------------ + --------------- - --------------------------- - --------------------------|*sin(2*x) 2 / -1 + sin(2*x)\ 60*|-sin(2*x) - ------------ + ------------------------ + ---------------------------| *|-1 + -------------| / -1 + sin(2*x)\ 4 / -1 + sin(2*x)\ / -1 + sin(2*x)\ / -1 + sin(2*x)\ 2 40*|-1 + -------------|*|-sin(2*x) - ------------ + ------------ + --------------- + ------------------------ - ---------------------------- - --------------------------- + --------------------------- + --------------------- - -------------------------------------| 80*|-sin(2*x) - ------------ + ------------------------ + ---------------------------|*|-1 + ------------- + ------------ + --------------- - --------------------------- - --------------------------| 96*|-1 + ------------- + ------------ + --------------- - --------------------------- - --------------------------|*sin(2*x) 2 / -1 + sin(2*x)\ 160*|-sin(2*x) - ------------ + ------------------------ + ---------------------------| 192*cos (2*x)*|-1 + ------------- + ------------ + --------------- - --------------------------- - --------------------------| 2 / -1 + sin(2*x)\ 560*cos (2*x)*|-1 + -------------|*|-sin(2*x) - ------------ + ------------------------ + ---------------------------| / -1 + sin(2*x)\ 2 200*|-1 + -------------| *cos (2*x)*|-sin(2*x) - ------------ + ------------------------ + ---------------------------| 200*|-1 + -------------|*|-sin(2*x) - ------------ + ------------------------ + ---------------------------|*sin(2*x) 192*cos (2*x)*|-1 + ------------- + ------------ + --------------- - --------------------------- - --------------------------| / -1 + sin(2*x)\ 4 2 / -1 + sin(2*x)\ / -1 + sin(2*x)\ 4 / -1 + sin(2*x)\ 2 / -1 + sin(2*x)\ 2 20*|-1 + -------------| *cos (2*x)*|-sin(2*x) - ------------ + ------------------------ + ---------------------------| / -1 + sin(2*x)\ 4 40*|-1 + -------------| *cos (2*x)*|-1 + ------------- + ------------ + --------------- - --------------------------- - --------------------------| 2 / -1 + sin(2*x)\ / -1 + sin(2*x)\ 4 200*cos (2*x)*|-1 + -------------|*|-1 + ------------- + ------------ + --------------- - --------------------------- - --------------------------| 384*|-sin(2*x) - ------------ + ------------------------ + ---------------------------|*sin(2*x) 4 / -1 + sin(2*x)\ 384*cos (2*x)*|-sin(2*x) - ------------ + ------------------------ + ---------------------------| / -1 + sin(2*x)\ 4 200*cos (2*x)*|-1 + -------------|*|-1 + ------------- + ------------ + --------------- - --------------------------- - --------------------------| 160*cos (2*x)*|-1 + -------------|*|-sin(2*x) - ------------ + ------------------------ + ---------------------------| / -1 + sin(2*x)\ 2 / -1 + sin(2*x)\ 2 200*|-1 + -------------| *cos (2*x)*|-sin(2*x) - ------------ + ------------------------ + ---------------------------| 200*|-1 + -------------|*|-sin(2*x) - ------------ + ------------------------ + ---------------------------|*sin(2*x) / -1 + sin(2*x)\ 2 2 / -1 + sin(2*x)\ 720*cos (2*x)*|-1 + -------------|*|-sin(2*x) - ------------ + ------------------------ + ---------------------------|| ___________________ | 2 | 1 + sin(2*x) 1 + sin(2*x) 2 | | 1 + sin(2*x) 1 + sin(2*x) 2 | | 1 + sin(2*x) 1 + sin(2*x) 3 1 + sin(2*x) 4 2 2 2 3 | | 1 + sin(2*x) 1 + sin(2*x) 2 | | 1 + sin(2*x) 1 + sin(2*x) 3 1 + sin(2*x) 4 2 2 2 3 | 2 4 |-1 + -------------| *cos (2*x) 4 2 400*|-1 + -------------| *cos (2*x) 384*cos (2*x)*|-1 + -------------| | 1 + sin(2*x) 1 + sin(2*x) 2 | | 1 + sin(2*x) 1 + sin(2*x) 2 | 140*|-1 + -------------| *cos (2*x) | 1 + sin(2*x) 1 + sin(2*x) 2 3 2 | 96*sin (2*x)*|-1 + -------------| | 1 + sin(2*x) 1 + sin(2*x) 2 | \ 1 + sin(2*x)/ 20*|-1 + -------------| *cos (2*x) 16*|-1 + -------------|*sin(2*x) 16*|-1 + -------------|*sin(2*x) 40*|-1 + -------------| *cos (2*x) \ 1 + sin(2*x)/ | 1 + sin(2*x) 1 + sin(2*x) 3 1 + sin(2*x) 4 2 2 2 3 | | 1 + sin(2*x) 1 + sin(2*x) 2 | | 1 + sin(2*x) 1 + sin(2*x) 2 3 2 | | 1 + sin(2*x) 1 + sin(2*x) 2 3 2 | 128*cos (2*x)*|-1 + -------------| | 1 + sin(2*x) 1 + sin(2*x) 2 | | 1 + sin(2*x) 1 + sin(2*x) 2 3 2 | 2 2 2 576*cos (2*x)*|-1 + -------------|*sin(2*x) \ 1 + sin(2*x)/ | 1 + sin(2*x) 1 + sin(2*x) 2 | 320*|-1 + -------------| *cos (2*x)*sin(2*x) \ 1 + sin(2*x)/ | 1 + sin(2*x) 1 + sin(2*x) 2 | \ 1 + sin(2*x)/ | 1 + sin(2*x) 1 + sin(2*x) 2 | | 1 + sin(2*x) 1 + sin(2*x) 2 3 2 | 160*|-1 + -------------| *cos (2*x) 128*cos (2*x)*|-1 + -------------| 60*|-1 + -------------| *cos (2*x) 40*|-1 + -------------| *cos (2*x) 40*|-1 + -------------| *cos (2*x)*sin(2*x) \ 1 + sin(2*x)/ | 1 + sin(2*x) 1 + sin(2*x) 2 | 20*|-1 + -------------| *cos (2*x) \ 1 + sin(2*x)/ | 1 + sin(2*x) 1 + sin(2*x) 2 3 2 | 96*sin (2*x)*|-1 + -------------| 200*|-1 + -------------| *cos (2*x) \ 1 + sin(2*x)/ | 1 + sin(2*x) 1 + sin(2*x) 2 3 2 | | 1 + sin(2*x) 1 + sin(2*x) 2 | 384*cos (2*x)*|-1 + -------------| | 1 + sin(2*x) 1 + sin(2*x) 2 | 560*|-1 + -------------| *cos (2*x) \ 1 + sin(2*x)/ | 1 + sin(2*x) 1 + sin(2*x) 2 3 2 | \ 1 + sin(2*x)/ | 1 + sin(2*x) 1 + sin(2*x) 2 | 80*|-1 + -------------| *cos (2*x)*sin(2*x) 40*|-1 + -------------| *cos (2*x)*sin(2*x) \ 1 + sin(2*x)/ | 1 + sin(2*x) 1 + sin(2*x) 2 | \ 1 + sin(2*x)/ | 1 + sin(2*x) 1 + sin(2*x) 2 | 400*|-1 + -------------| *cos (2*x)*sin(2*x) 576*cos (2*x)*|-1 + -------------|*sin(2*x) \ 1 + sin(2*x)/ | 1 + sin(2*x) 1 + sin(2*x) 2 || / -(-1 + sin(2*x)) | 960*cos (2*x) 480*sin(2*x) \ (1 + sin(2*x)) / \ (1 + sin(2*x)) / \ (1 + sin(2*x)) (1 + sin(2*x)) (1 + sin(2*x)) (1 + sin(2*x)) (1 + sin(2*x)) (1 + sin(2*x)) / 16*(-1 + sin(2*x)) \ (1 + sin(2*x)) / \ (1 + sin(2*x)) (1 + sin(2*x)) (1 + sin(2*x)) (1 + sin(2*x)) (1 + sin(2*x)) (1 + sin(2*x)) / 1440*sin (2*x) 1920*cos (2*x) \ 1 + sin(2*x)/ 1920*cos (2*x)*(-1 + sin(2*x)) 1440*sin (2*x)*(-1 + sin(2*x)) \ 1 + sin(2*x)/ \ 1 + sin(2*x)/ \ (1 + sin(2*x)) / \ (1 + sin(2*x)) / \ 1 + sin(2*x)/ \ (1 + sin(2*x)) (1 + sin(2*x)) (1 + sin(2*x)) / \ 1 + sin(2*x)/ \ (1 + sin(2*x)) / \ 1 + sin(2*x)/ \ 1 + sin(2*x)/ \ 1 + sin(2*x)/ \ 1 + sin(2*x)/ \ (1 + sin(2*x)) (1 + sin(2*x)) (1 + sin(2*x)) (1 + sin(2*x)) (1 + sin(2*x)) (1 + sin(2*x)) / \ (1 + sin(2*x)) / \ (1 + sin(2*x)) (1 + sin(2*x)) (1 + sin(2*x)) / \ (1 + sin(2*x)) (1 + sin(2*x)) (1 + sin(2*x)) / \ 1 + sin(2*x)/ \ (1 + sin(2*x)) / \ (1 + sin(2*x)) (1 + sin(2*x)) (1 + sin(2*x)) / 480*(-1 + sin(2*x))*sin(2*x) 960*cos (2*x)*(-1 + sin(2*x)) 3840*cos (2*x)*sin(2*x) 3840*cos (2*x)*(-1 + sin(2*x))*sin(2*x) \ 1 + sin(2*x)/ \ (1 + sin(2*x)) / \ 1 + sin(2*x)/ \ (1 + sin(2*x)) / \ (1 + sin(2*x)) / \ (1 + sin(2*x)) (1 + sin(2*x)) (1 + sin(2*x)) / \ 1 + sin(2*x)/ \ 1 + sin(2*x)/ \ 1 + sin(2*x)/ \ 1 + sin(2*x)/ \ 1 + sin(2*x)/ \ (1 + sin(2*x)) / \ 1 + sin(2*x)/ \ (1 + sin(2*x)) (1 + sin(2*x)) (1 + sin(2*x)) / \ 1 + sin(2*x)/ \ 1 + sin(2*x)/ \ (1 + sin(2*x)) (1 + sin(2*x)) (1 + sin(2*x)) / \ (1 + sin(2*x)) / \ 1 + sin(2*x)/ \ (1 + sin(2*x)) / \ 1 + sin(2*x)/ \ (1 + sin(2*x)) (1 + sin(2*x)) (1 + sin(2*x)) / \ (1 + sin(2*x)) / \ 1 + sin(2*x)/ \ 1 + sin(2*x)/ \ (1 + sin(2*x)) / \ (1 + sin(2*x)) / \ 1 + sin(2*x)/ \ 1 + sin(2*x)/ \ (1 + sin(2*x)) /| / ----------------- *|16 - --------------- - ------------ - ---------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------ + -------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + --------------- + --------------- - ------------------------------- - ------------------------------ - ------------------------------ - ----------------------------------- - ---------------------------------- - ------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------ - ----------------------------------- - ---------------------------------------------------------------------------------------------------------------------------- - --------------------------------- - ------------------------------------------------------------------------------------------------------------ - ---------------------------------- - -------------------------------- + -------------------------------- + ---------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------- + ---------------------------------- + ---------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------ + ---------------------------- + ----------------------------- + ----------------------- - --------------------------------------- - ------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------- - -------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------ - ----------------------------------- - ---------------------------------- - ---------------------------------- - ---------------------------------- - ------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------- + ---------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------------------- + --------------------------------- + ----------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------ + ---------------------------------- + ------------------------------------------------------------------------------------------------- + ----------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------- - ------------------------------------------- + ------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------- + -------------------------------------------- + ------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------|*cos(2*x) \/ 1 + sin(2*x) | 2 1 + sin(2*x) 2 1 + sin(2*x) 1 + sin(2*x) 1 + sin(2*x) -1 + sin(2*x) -1 + sin(2*x) 2 4 4 5 3 4 4 3 2 4 1 + sin(2*x) 2 2 4 1 + sin(2*x) -1 + sin(2*x) 2 -1 + sin(2*x) -1 + sin(2*x) -1 + sin(2*x) 2 (1 + sin(2*x))*(-1 + sin(2*x)) 2 2 3 3 4 3 3 3 3 2 (1 + sin(2*x))*(-1 + sin(2*x)) 2 2 (1 + sin(2*x))*(-1 + sin(2*x)) 2 2 (1 + sin(2*x))*(-1 + sin(2*x)) 3 3 3 2 (1 + sin(2*x))*(-1 + sin(2*x)) 3 2 (1 + sin(2*x))*(-1 + sin(2*x)) 3 2 3 (1 + sin(2*x))*(-1 + sin(2*x)) 2 2 2 2 (1 + sin(2*x))*(-1 + sin(2*x)) 2 2 2 | \ (1 + sin(2*x)) (-1 + sin(2*x)) (1 + sin(2*x)) (1 + sin(2*x)) (-1 + sin(2*x)) (1 + sin(2*x)) (1 + sin(2*x)) (-1 + sin(2*x)) (-1 + sin(2*x)) (-1 + sin(2*x)) (-1 + sin(2*x)) (-1 + sin(2*x)) (-1 + sin(2*x)) (-1 + sin(2*x)) (-1 + sin(2*x)) (-1 + sin(2*x)) (-1 + sin(2*x)) (-1 + sin(2*x)) (1 + sin(2*x)) (1 + sin(2*x)) (1 + sin(2*x)) (1 + sin(2*x)) (-1 + sin(2*x)) (-1 + sin(2*x)) (-1 + sin(2*x)) (-1 + sin(2*x)) (-1 + sin(2*x)) (1 + sin(2*x)) *(-1 + sin(2*x)) (1 + sin(2*x)) *(-1 + sin(2*x)) (-1 + sin(2*x)) (-1 + sin(2*x)) (1 + sin(2*x))*(-1 + sin(2*x)) (-1 + sin(2*x)) (1 + sin(2*x))*(-1 + sin(2*x)) (-1 + sin(2*x)) (1 + sin(2*x))*(-1 + sin(2*x)) (1 + sin(2*x))*(-1 + sin(2*x)) (1 + sin(2*x))*(-1 + sin(2*x)) (1 + sin(2*x)) *(-1 + sin(2*x)) (1 + sin(2*x)) *(-1 + sin(2*x)) (1 + sin(2*x))*(-1 + sin(2*x)) (1 + sin(2*x))*(-1 + sin(2*x)) (1 + sin(2*x))*(-1 + sin(2*x)) (1 + sin(2*x))*(-1 + sin(2*x)) (1 + sin(2*x))*(-1 + sin(2*x)) / ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -1 + sin(2*x)