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Similar expressions
ctg((x-1)/sqrt(2))

The derivative of the function/ ctg((x+1)/sqrt(2))

Derivative of ctg((x+1)/sqrt(2))

The solution

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   /x + 1\
cot|-----|
   |  ___|
   \\/ 2 /

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

cot((x + 1)/sqrt(2))

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

       /        /  ___        \\ /          /  ___        \\ 
   ___ |       2|\/ 2 *(1 + x)|| |         2|\/ 2 *(1 + x)|| 
-\/ 2 *|1 + cot |-------------||*|1 + 3*cot |-------------|| 
       \        \      2      // \          \      2      // 
-------------------------------------------------------------
                              2

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

Simplify

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Derivative of ctg((x+1)/sqrt(2))

The graph:

Piecewise:

The solution

Method #1

Method #2