Derivative of x/(1-cos2x)

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     x      
------------
1 - cos(2*x)

$$\frac{x}{1 - \cos{\left(2 x \right)}}$$

x/(1 - cos(2*x))

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $1 - \cos{\left(2 x \right)}$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Let $u = 2 x$ .
    2. The derivative of cosine is negative sine:
      
      $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
      1. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $2$
      The result of the chain rule is:
      
      $- 2 \sin{\left(2 x \right)}$
    So, the result is: $2 \sin{\left(2 x \right)}$
  The result is: $2 \sin{\left(2 x \right)}$
Now plug in to the quotient rule:

$\frac{- 2 x \sin{\left(2 x \right)} - \cos{\left(2 x \right)} + 1}{\left(1 - \cos{\left(2 x \right)}\right)^{2}}$
Now simplify:

$\frac{- \frac{x}{\tan{\left(x \right)}} + \frac{1}{2}}{\sin^{2}{\left(x \right)}}$

The answer is:

$\frac{- \frac{x}{\tan{\left(x \right)}} + \frac{1}{2}}{\sin^{2}{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     1           2*x*sin(2*x) 
------------ - ---------------
1 - cos(2*x)                 2
               (1 - cos(2*x))

$$- \frac{2 x \sin{\left(2 x \right)}}{\left(1 - \cos{\left(2 x \right)}\right)^{2}} + \frac{1}{1 - \cos{\left(2 x \right)}}$$

Simplify

The second derivative [src]

   /  /      2                 \           \
   |  | 2*sin (2*x)            |           |
-4*|x*|------------- + cos(2*x)| + sin(2*x)|
   \  \-1 + cos(2*x)           /           /
--------------------------------------------
                             2              
              (-1 + cos(2*x))

$$- \frac{4 \left(x \left(\cos{\left(2 x \right)} + \frac{2 \sin^{2}{\left(2 x \right)}}{\cos{\left(2 x \right)} - 1}\right) + \sin{\left(2 x \right)}\right)}{\left(\cos{\left(2 x \right)} - 1\right)^{2}}$$

Simplify

The third derivative [src]

   /                   2             /                            2        \         \
   |              6*sin (2*x)        |       6*cos(2*x)      6*sin (2*x)   |         |
-4*|3*cos(2*x) + ------------- + 2*x*|-1 + ------------- + ----------------|*sin(2*x)|
   |             -1 + cos(2*x)       |     -1 + cos(2*x)                  2|         |
   \                                 \                     (-1 + cos(2*x)) /         /
--------------------------------------------------------------------------------------
                                                  2                                   
                                   (-1 + cos(2*x))

$$- \frac{4 \left(2 x \left(-1 + \frac{6 \cos{\left(2 x \right)}}{\cos{\left(2 x \right)} - 1} + \frac{6 \sin^{2}{\left(2 x \right)}}{\left(\cos{\left(2 x \right)} - 1\right)^{2}}\right) \sin{\left(2 x \right)} + 3 \cos{\left(2 x \right)} + \frac{6 \sin^{2}{\left(2 x \right)}}{\cos{\left(2 x \right)} - 1}\right)}{\left(\cos{\left(2 x \right)} - 1\right)^{2}}$$

Simplify

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Derivative of x/(1-cos2x)

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