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Derivative of (1-2sin2x)\(cos6x)

Function f() - derivative -N order at the point
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The solution

You have entered [src]
1 - 2*sin(2*x)
--------------
   cos(6*x)   
$$\frac{1 - 2 \sin{\left(2 x \right)}}{\cos{\left(6 x \right)}}$$
(1 - 2*sin(2*x))/cos(6*x)
Detail solution
  1. Apply the quotient rule, which is:

    and .

    To find :

    1. Differentiate term by term:

      1. The derivative of the constant is zero.

      2. The derivative of a constant times a function is the constant times the derivative of the function.

        1. Let .

        2. The derivative of sine is cosine:

        3. Then, apply the chain rule. Multiply by :

          1. The derivative of a constant times a function is the constant times the derivative of the function.

            1. 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 :

    1. Let .

    2. The derivative of cosine is negative sine:

    3. Then, apply the chain rule. Multiply by :

      1. The derivative of a constant times a function is the constant times the derivative of the function.

        1. Apply the power rule: goes to

        So, the result is:

      The result of the chain rule is:

    Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
  4*cos(2*x)   6*(1 - 2*sin(2*x))*sin(6*x)
- ---------- + ---------------------------
   cos(6*x)                2              
                        cos (6*x)         
$$\frac{6 \left(1 - 2 \sin{\left(2 x \right)}\right) \sin{\left(6 x \right)}}{\cos^{2}{\left(6 x \right)}} - \frac{4 \cos{\left(2 x \right)}}{\cos{\left(6 x \right)}}$$
The second derivative [src]
  /               /         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)                                  
$$\frac{4 \left(- 9 \left(\frac{2 \sin^{2}{\left(6 x \right)}}{\cos^{2}{\left(6 x \right)}} + 1\right) \left(2 \sin{\left(2 x \right)} - 1\right) + 2 \sin{\left(2 x \right)} - \frac{12 \sin{\left(6 x \right)} \cos{\left(2 x \right)}}{\cos{\left(6 x \right)}}\right)}{\cos{\left(6 x \right)}}$$
The third derivative [src]
  /                                                                                         /         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)                                                       
$$\frac{8 \left(- 54 \left(\frac{2 \sin^{2}{\left(6 x \right)}}{\cos^{2}{\left(6 x \right)}} + 1\right) \cos{\left(2 x \right)} - \frac{27 \left(\frac{6 \sin^{2}{\left(6 x \right)}}{\cos^{2}{\left(6 x \right)}} + 5\right) \left(2 \sin{\left(2 x \right)} - 1\right) \sin{\left(6 x \right)}}{\cos{\left(6 x \right)}} + \frac{18 \sin{\left(2 x \right)} \sin{\left(6 x \right)}}{\cos{\left(6 x \right)}} + 2 \cos{\left(2 x \right)}\right)}{\cos{\left(6 x \right)}}$$