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Identical expressions
(z- two pi)^2/(cos(z)- one)
(z minus 2 Pi ) squared divide by ( co sinus of e of (z) minus 1)
(z minus two Pi ) squared divide by ( co sinus of e of (z) minus one)
(z-2pi)2/(cos(z)-1)
z-2pi2/cosz-1
(z-2pi)²/(cos(z)-1)
(z-2pi) to the power of 2/(cos(z)-1)
z-2pi^2/cosz-1
(z-2pi)^2 divide by (cos(z)-1)
Similar expressions
(z+2pi)^2/(cos(z)-1)
(z-2pi)^2/(cos(z)+1)

The derivative of the function/ (z-2pi)^2/(cos(z)-1)

Derivative of (z-2pi)^2/(cos(z)-1)

The solution

You have entered [src]

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

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

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

Detail solution

Apply the quotient rule, which is:

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

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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                        2       
-4*pi + 2*z   (z - 2*pi) *sin(z)
----------- + ------------------
 cos(z) - 1                 2   
                (cos(z) - 1)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of (z-2pi)^2/(cos(z)-1)

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The solution