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Identical expressions
(z/(one -cosz))(z- two *pi*k)²
(z divide by (1 minus co sinus of e of z))(z minus 2 multiply by Pi multiply by k)²
(z divide by (one minus co sinus of e of z))(z minus two multiply by Pi multiply by k)²
(z/(1-cosz))(z-2pik)²
z/1-coszz-2pik²
(z divide by (1-cosz))(z-2*pi*k)²
Similar expressions
(z/(1-cosz))(z+2*pi*k)²
(z/(1+cosz))(z-2*pi*k)²

The derivative of the function/ (z/(1-cosz))(z-2*pi*k)²

Derivative of (z/(1-cosz))(z-2pik)²

The solution

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    z                  2
----------*(z - 2*pi*k) 
1 - cos(z)

$$\frac{z}{1 - \cos{\left(z \right)}} \left(- 2 \pi k + z\right)^{2}$$

(z/(1 - cos(z)))*(z - 2*pi*k)^2

Detail solution

Apply the quotient rule, which is:

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

$f{\left(z \right)} = z \left(- 2 \pi k + z\right)^{2}$ and $g{\left(z \right)} = 1 - \cos{\left(z \right)}$ .

To find $\frac{d}{d z} f{\left(z \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d z} f{\left(z \right)} g{\left(z \right)} = f{\left(z \right)} \frac{d}{d z} g{\left(z \right)} + g{\left(z \right)} \frac{d}{d z} f{\left(z \right)}$
  
  $f{\left(z \right)} = z$ ; to find $\frac{d}{d z} f{\left(z \right)}$ :
  1. Apply the power rule: $z$ goes to $1$
  $g{\left(z \right)} = \left(- 2 \pi k + z\right)^{2}$ ; to find $\frac{d}{d z} g{\left(z \right)}$ :
  1. Let $u = - 2 \pi k + z$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  3. Then, apply the chain rule. Multiply by $\frac{\partial}{\partial z} \left(- 2 \pi k + z\right)$ :
    1. Differentiate $- 2 \pi k + z$ term by term:
      1. Apply the power rule: $z$ goes to $1$
      2. The derivative of the constant $- 2 \pi k$ is zero.
      The result is: $1$
    The result of the chain rule is:
    
    $- 4 \pi k + 2 z$
  The result is: $z \left(- 4 \pi k + 2 z\right) + \left(- 2 \pi k + z\right)^{2}$
To find $\frac{d}{d z} g{\left(z \right)}$ :
1. Differentiate $1 - \cos{\left(z \right)}$ 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. The derivative of cosine is negative sine:
      
      $\frac{d}{d z} \cos{\left(z \right)} = - \sin{\left(z \right)}$
    So, the result is: $\sin{\left(z \right)}$
  The result is: $\sin{\left(z \right)}$
Now plug in to the quotient rule:

$\frac{- z \left(- 2 \pi k + z\right)^{2} \sin{\left(z \right)} + \left(1 - \cos{\left(z \right)}\right) \left(z \left(- 4 \pi k + 2 z\right) + \left(- 2 \pi k + z\right)^{2}\right)}{\left(1 - \cos{\left(z \right)}\right)^{2}}$
Now simplify:

$\frac{\left(2 \pi k - z\right) \left(- z \left(2 \pi k - z\right) \sin{\left(z \right)} + \left(- 2 \pi k + 3 z\right) \left(\cos{\left(z \right)} - 1\right)\right)}{\left(\cos{\left(z \right)} - 1\right)^{2}}$

The answer is:

$\frac{\left(2 \pi k - z\right) \left(- z \left(2 \pi k - z\right) \sin{\left(z \right)} + \left(- 2 \pi k + 3 z\right) \left(\cos{\left(z \right)} - 1\right)\right)}{\left(\cos{\left(z \right)} - 1\right)^{2}}$

The first derivative [src]

            2 /    1           z*sin(z)  \   z*(2*z - 4*pi*k)
(z - 2*pi*k) *|---------- - -------------| + ----------------
              |1 - cos(z)               2|      1 - cos(z)   
              \             (1 - cos(z)) /

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

Simplify

The second derivative [src]

                                                          /             /      2             \\
                                                        2 |             | 2*sin (z)          ||
                                           (-z + 2*pi*k) *|2*sin(z) + z*|----------- + cos(z)||
         /      z*sin(z) \                                \             \-1 + cos(z)         //
-2*z + 4*|1 + -----------|*(-z + 2*pi*k) - ----------------------------------------------------
         \    -1 + cos(z)/                                     -1 + cos(z)                     
-----------------------------------------------------------------------------------------------
                                          -1 + cos(z)

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

Simplify

The third derivative [src]

                    /                 2         /                          2      \       \                                                                      
                  2 |            6*sin (z)      |       6*cos(z)      6*sin (z)   |       |                                 /             /      2             \\
     (-z + 2*pi*k) *|3*cos(z) + ----------- + z*|-1 + ----------- + --------------|*sin(z)|                                 |             | 2*sin (z)          ||
                    |           -1 + cos(z)     |     -1 + cos(z)                2|       |                 6*(-z + 2*pi*k)*|2*sin(z) + z*|----------- + cos(z)||
                    \                           \                   (-1 + cos(z)) /       /    6*z*sin(z)                   \             \-1 + cos(z)         //
-6 - -------------------------------------------------------------------------------------- - ----------- + -----------------------------------------------------
                                          -1 + cos(z)                                         -1 + cos(z)                        -1 + cos(z)                     
-----------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                           -1 + cos(z)

$$\frac{- \frac{6 z \sin{\left(z \right)}}{\cos{\left(z \right)} - 1} - \frac{\left(2 \pi k - z\right)^{2} \left(z \left(-1 + \frac{6 \cos{\left(z \right)}}{\cos{\left(z \right)} - 1} + \frac{6 \sin^{2}{\left(z \right)}}{\left(\cos{\left(z \right)} - 1\right)^{2}}\right) \sin{\left(z \right)} + 3 \cos{\left(z \right)} + \frac{6 \sin^{2}{\left(z \right)}}{\cos{\left(z \right)} - 1}\right)}{\cos{\left(z \right)} - 1} + \frac{6 \left(2 \pi k - z\right) \left(z \left(\cos{\left(z \right)} + \frac{2 \sin^{2}{\left(z \right)}}{\cos{\left(z \right)} - 1}\right) + 2 \sin{\left(z \right)}\right)}{\cos{\left(z \right)} - 1} - 6}{\cos{\left(z \right)} - 1}$$

Simplify

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Derivative of (z/(1-cosz))(z-2pik)²

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The solution

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Derivative of (z/(1-cosz))(z-2*pi*k)²

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Piecewise:

The solution

Derivative of (z/(1-cosz))(z-2pik)²