Mister Exam

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)   
12sin(2x)cos(6x)\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:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)g2(x)\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}

    f(x)=12sin(2x)f{\left(x \right)} = 1 - 2 \sin{\left(2 x \right)} and g(x)=cos(6x)g{\left(x \right)} = \cos{\left(6 x \right)}.

    To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Differentiate 12sin(2x)1 - 2 \sin{\left(2 x \right)} term by term:

      1. The derivative of the constant 11 is zero.

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

        1. Let u=2xu = 2 x.

        2. The derivative of sine is cosine:

          ddusin(u)=cos(u)\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}

        3. Then, apply the chain rule. Multiply by ddx2x\frac{d}{d x} 2 x:

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

            1. Apply the power rule: xx goes to 11

            So, the result is: 22

          The result of the chain rule is:

          2cos(2x)2 \cos{\left(2 x \right)}

        So, the result is: 4cos(2x)- 4 \cos{\left(2 x \right)}

      The result is: 4cos(2x)- 4 \cos{\left(2 x \right)}

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Let u=6xu = 6 x.

    2. The derivative of cosine is negative sine:

      dducos(u)=sin(u)\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}

    3. Then, apply the chain rule. Multiply by ddx6x\frac{d}{d x} 6 x:

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

        1. Apply the power rule: xx goes to 11

        So, the result is: 66

      The result of the chain rule is:

      6sin(6x)- 6 \sin{\left(6 x \right)}

    Now plug in to the quotient rule:

    6(12sin(2x))sin(6x)4cos(2x)cos(6x)cos2(6x)\frac{6 \left(1 - 2 \sin{\left(2 x \right)}\right) \sin{\left(6 x \right)} - 4 \cos{\left(2 x \right)} \cos{\left(6 x \right)}}{\cos^{2}{\left(6 x \right)}}

  2. Now simplify:

    2(3sin(6x)4cos(4x)+2cos(8x))cos2(6x)\frac{2 \left(3 \sin{\left(6 x \right)} - 4 \cos{\left(4 x \right)} + 2 \cos{\left(8 x \right)}\right)}{\cos^{2}{\left(6 x \right)}}


The answer is:

2(3sin(6x)4cos(4x)+2cos(8x))cos2(6x)\frac{2 \left(3 \sin{\left(6 x \right)} - 4 \cos{\left(4 x \right)} + 2 \cos{\left(8 x \right)}\right)}{\cos^{2}{\left(6 x \right)}}

The graph
02468-8-6-4-2-1010-2500025000
The first derivative [src]
  4*cos(2*x)   6*(1 - 2*sin(2*x))*sin(6*x)
- ---------- + ---------------------------
   cos(6*x)                2              
                        cos (6*x)         
6(12sin(2x))sin(6x)cos2(6x)4cos(2x)cos(6x)\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)                                  
4(9(2sin2(6x)cos2(6x)+1)(2sin(2x)1)+2sin(2x)12sin(6x)cos(2x)cos(6x))cos(6x)\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)                                                       
8(54(2sin2(6x)cos2(6x)+1)cos(2x)27(6sin2(6x)cos2(6x)+5)(2sin(2x)1)sin(6x)cos(6x)+18sin(2x)sin(6x)cos(6x)+2cos(2x))cos(6x)\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)}}