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Derivative of:
Derivative of x^4*e^x
Derivative of x^3+1/x-1
Derivative of sin(x)+cot(x)
Derivative of e^x-2
Identical expressions
ctg(pix/ two)^- one
ctg( Pi x divide by 2) to the power of minus 1
ctg( Pi x divide by two) to the power of minus one
ctg(pix/2)-1
ctgpix/2-1
ctgpix/2^-1
ctg(pix divide by 2)^-1
Similar expressions
ctg(pix/2)^+1

The derivative of the function/ ctg(pix/2)^-1

Derivative of ctg(pix/2)^-1

The solution

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    1    
---------
   /pi*x\
cot|----|
   \ 2  /

$$\frac{1}{\cot{\left(\frac{\pi x}{2} \right)}}$$

1/cot((pi*x)/2)

Detail solution

Let $u = \cot{\left(\frac{\pi x}{2} \right)}$ .
Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \cot{\left(\frac{\pi x}{2} \right)}$ :
1. There are multiple ways to do this derivative.
  Method #1
  1. Rewrite the function to be differentiated:
    
    $\cot{\left(\frac{\pi x}{2} \right)} = \frac{1}{\tan{\left(\frac{\pi x}{2} \right)}}$
  2. Let $u = \tan{\left(\frac{\pi x}{2} \right)}$ .
  3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
  4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(\frac{\pi x}{2} \right)}$ :
    1. Rewrite the function to be differentiated:
      
      $\tan{\left(\frac{\pi x}{2} \right)} = \frac{\sin{\left(\frac{\pi x}{2} \right)}}{\cos{\left(\frac{\pi x}{2} \right)}}$
    2. Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \sin{\left(\frac{\pi x}{2} \right)}$ and $g{\left(x \right)} = \cos{\left(\frac{\pi x}{2} \right)}$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      
      Let $u = \frac{\pi x}{2}$ .
      
      The derivative of sine is cosine:
      
      $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{\pi x}{2}$ :
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      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: $\pi$
      
      So, the result is: $\frac{\pi}{2}$
      
      The result of the chain rule is:
      
      $\frac{\pi \cos{\left(\frac{\pi x}{2} \right)}}{2}$
      
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      
      Let $u = \frac{\pi x}{2}$ .
      
      The derivative of cosine is negative sine:
      
      $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{\pi x}{2}$ :
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      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: $\pi$
      
      So, the result is: $\frac{\pi}{2}$
      
      The result of the chain rule is:
      
      $- \frac{\pi \sin{\left(\frac{\pi x}{2} \right)}}{2}$
      
      Now plug in to the quotient rule:
      
      $\frac{\frac{\pi \sin^{2}{\left(\frac{\pi x}{2} \right)}}{2} + \frac{\pi \cos^{2}{\left(\frac{\pi x}{2} \right)}}{2}}{\cos^{2}{\left(\frac{\pi x}{2} \right)}}$
    The result of the chain rule is:
    
    $- \frac{\frac{\pi \sin^{2}{\left(\frac{\pi x}{2} \right)}}{2} + \frac{\pi \cos^{2}{\left(\frac{\pi x}{2} \right)}}{2}}{\cos^{2}{\left(\frac{\pi x}{2} \right)} \tan^{2}{\left(\frac{\pi x}{2} \right)}}$
  Method #2
  1. Rewrite the function to be differentiated:
    
    $\cot{\left(\frac{\pi x}{2} \right)} = \frac{\cos{\left(\frac{\pi x}{2} \right)}}{\sin{\left(\frac{\pi x}{2} \right)}}$
  2. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \cos{\left(\frac{\pi x}{2} \right)}$ and $g{\left(x \right)} = \sin{\left(\frac{\pi x}{2} \right)}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Let $u = \frac{\pi x}{2}$ .
    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 $\frac{d}{d x} \frac{\pi x}{2}$ :
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      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: $\pi$
      
      So, the result is: $\frac{\pi}{2}$
      
      The result of the chain rule is:
      
      $- \frac{\pi \sin{\left(\frac{\pi x}{2} \right)}}{2}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Let $u = \frac{\pi x}{2}$ .
    2. The derivative of sine is cosine:
      
      $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{\pi x}{2}$ :
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      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: $\pi$
      
      So, the result is: $\frac{\pi}{2}$
      
      The result of the chain rule is:
      
      $\frac{\pi \cos{\left(\frac{\pi x}{2} \right)}}{2}$
    Now plug in to the quotient rule:
    
    $\frac{- \frac{\pi \sin^{2}{\left(\frac{\pi x}{2} \right)}}{2} - \frac{\pi \cos^{2}{\left(\frac{\pi x}{2} \right)}}{2}}{\sin^{2}{\left(\frac{\pi x}{2} \right)}}$
The result of the chain rule is:

$\frac{\frac{\pi \sin^{2}{\left(\frac{\pi x}{2} \right)}}{2} + \frac{\pi \cos^{2}{\left(\frac{\pi x}{2} \right)}}{2}}{\cos^{2}{\left(\frac{\pi x}{2} \right)} \tan^{2}{\left(\frac{\pi x}{2} \right)} \cot^{2}{\left(\frac{\pi x}{2} \right)}}$
Now simplify:

$\frac{\pi}{\cos{\left(\pi x \right)} + 1}$

The answer is:

$\frac{\pi}{\cos{\left(\pi x \right)} + 1}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    /        2/pi*x\\ 
-pi*|-1 - cot |----|| 
    \         \ 2  // 
----------------------
          2/pi*x\     
     2*cot |----|     
           \ 2  /

$$- \frac{\pi \left(- \cot^{2}{\left(\frac{\pi x}{2} \right)} - 1\right)}{2 \cot^{2}{\left(\frac{\pi x}{2} \right)}}$$

Simplify

The second derivative [src]

                     /            2/pi*x\\
                     |     1 + cot |----||
  2 /       2/pi*x\\ |             \ 2  /|
pi *|1 + cot |----||*|-1 + --------------|
    \        \ 2  // |          2/pi*x\  |
                     |       cot |----|  |
                     \           \ 2  /  /
------------------------------------------
                    /pi*x\                
               2*cot|----|                
                    \ 2  /

$$\frac{\pi^{2} \left(\frac{\cot^{2}{\left(\frac{\pi x}{2} \right)} + 1}{\cot^{2}{\left(\frac{\pi x}{2} \right)}} - 1\right) \left(\cot^{2}{\left(\frac{\pi x}{2} \right)} + 1\right)}{2 \cot{\left(\frac{\pi x}{2} \right)}}$$

Simplify

The third derivative [src]

                     /                                           2\
                     |      /       2/pi*x\\     /       2/pi*x\\ |
                     |    5*|1 + cot |----||   3*|1 + cot |----|| |
  3 /       2/pi*x\\ |      \        \ 2  //     \        \ 2  // |
pi *|1 + cot |----||*|2 - ------------------ + -------------------|
    \        \ 2  // |           2/pi*x\               4/pi*x\    |
                     |        cot |----|            cot |----|    |
                     \            \ 2  /                \ 2  /    /
-------------------------------------------------------------------
                                 4

$$\frac{\pi^{3} \left(\cot^{2}{\left(\frac{\pi x}{2} \right)} + 1\right) \left(\frac{3 \left(\cot^{2}{\left(\frac{\pi x}{2} \right)} + 1\right)^{2}}{\cot^{4}{\left(\frac{\pi x}{2} \right)}} - \frac{5 \left(\cot^{2}{\left(\frac{\pi x}{2} \right)} + 1\right)}{\cot^{2}{\left(\frac{\pi x}{2} \right)}} + 2\right)}{4}$$

Simplify

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Derivative of ctg(pix/2)^-1

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Piecewise:

The solution

Method #1

Method #2