1 - 2*sin(2*x) -------------- cos(6*x)
(1 - 2*sin(2*x))/cos(6*x)
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 :
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:
Now plug in to the quotient rule:
Now simplify:
The answer is:
4*cos(2*x) 6*(1 - 2*sin(2*x))*sin(6*x) - ---------- + --------------------------- cos(6*x) 2 cos (6*x)
/ / 2 \ \ | | 2*sin (6*x)| 12*cos(2*x)*sin(6*x)| 4*|2*sin(2*x) - 9*|1 + -----------|*(-1 + 2*sin(2*x)) - --------------------| | | 2 | cos(6*x) | \ \ cos (6*x) / / ----------------------------------------------------------------------------- cos(6*x)
/ / 2 \ \ | | 6*sin (6*x)| | | 27*(-1 + 2*sin(2*x))*|5 + -----------|*sin(6*x)| | / 2 \ | 2 | | | | 2*sin (6*x)| 18*sin(2*x)*sin(6*x) \ cos (6*x) / | 8*|2*cos(2*x) - 54*|1 + -----------|*cos(2*x) + -------------------- - -----------------------------------------------| | | 2 | cos(6*x) cos(6*x) | \ \ cos (6*x) / / ----------------------------------------------------------------------------------------------------------------------- cos(6*x)