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Derivative of:
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Derivative of x^-(4/5)
Derivative of (x^2+49)/x
Derivative of y=2x-1
Identical expressions
(ctg(pix/ two))/(ln(x- two))
(ctg( Pi x divide by 2)) divide by (ln(x minus 2))
(ctg( Pi x divide by two)) divide by (ln(x minus two))
ctgpix/2/lnx-2
(ctg(pix divide by 2)) divide by (ln(x-2))
Similar expressions
(ctg(pix/2))/(ln(x+2))

The derivative of the function/ (ctg(pix/2))/(ln(x-2))

Derivative of (ctg(pix/2))/(ln(x-2))

The solution

You have entered [src]

   /pi*x\ 
cot|----| 
   \ 2  / 
----------
log(x - 2)

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

cot((pi*x)/2)/log(x - 2)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \cot{\left(\frac{\pi x}{2} \right)}$ and $g{\left(x \right)} = \log{\left(x - 2 \right)}$ .

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

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

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

The answer is:

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

The first derivative [src]

   /        2/pi*x\\           /pi*x\     
pi*|-1 - cot |----||        cot|----|     
   \         \ 2  //           \ 2  /     
-------------------- - -------------------
    2*log(x - 2)                  2       
                       (x - 2)*log (x - 2)

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

Simplify

The second derivative [src]

  2 /       2/pi*x\\    /pi*x\      /       2/pi*x\\    /         2     \    /pi*x\
pi *|1 + cot |----||*cot|----|   pi*|1 + cot |----||    |1 + -----------|*cot|----|
    \        \ 2  //    \ 2  /      \        \ 2  //    \    log(-2 + x)/    \ 2  /
------------------------------ + -------------------- + ---------------------------
              2                  (-2 + x)*log(-2 + x)              2               
                                                           (-2 + x) *log(-2 + x)   
-----------------------------------------------------------------------------------
                                    log(-2 + x)

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

Simplify

The third derivative [src]

 /                                            /         3             3      \    /pi*x\                                                                             \ 
 |  3 /       2/pi*x\\ /         2/pi*x\\   2*|1 + ----------- + ------------|*cot|----|        /       2/pi*x\\ /         2     \       2 /       2/pi*x\\    /pi*x\| 
 |pi *|1 + cot |----||*|1 + 3*cot |----||     |    log(-2 + x)      2        |    \ 2  /   3*pi*|1 + cot |----||*|1 + -----------|   3*pi *|1 + cot |----||*cot|----|| 
 |    \        \ 2  // \          \ 2  //     \                  log (-2 + x)/                  \        \ 2  // \    log(-2 + x)/         \        \ 2  //    \ 2  /| 
-|--------------------------------------- + -------------------------------------------- + --------------------------------------- + --------------------------------| 
 |                   4                                         3                                             2                            2*(-2 + x)*log(-2 + x)     | 
 \                                                     (-2 + x) *log(-2 + x)                       2*(-2 + x) *log(-2 + x)                                           / 
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                              log(-2 + x)

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

Simplify

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Derivative of (ctg(pix/2))/(ln(x-2))

The graph:

Piecewise:

The solution

Method #1

Method #2