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Derivative of:
Derivative of e^-1
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Identical expressions
(log(cot(x/ two)+ one))
( logarithm of ( cotangent of (x divide by 2) plus 1))
( logarithm of ( cotangent of (x divide by two) plus one))
logcotx/2+1
(log(cot(x divide by 2)+1))
Similar expressions
(log(cot(x/2)-1))

The derivative of the function/ (log(cot(x/2)+1))

Derivative of (log(cot(x/2)+1))

The solution

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

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

log(cot(x/2) + 1)

Detail solution

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

$- \frac{\frac{\sin^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{2}}{\left(\cot{\left(\frac{x}{2} \right)} + 1\right) \cos^{2}{\left(\frac{x}{2} \right)} \tan^{2}{\left(\frac{x}{2} \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

              /                  2/x\\
              |           1 + cot |-||
/       2/x\\ |     /x\           \2/|
|1 + cot |-||*|2*cot|-| - -----------|
\        \2// |     \2/           /x\|
              |            1 + cot|-||
              \                   \2//
--------------------------------------
              /       /x\\            
            4*|1 + cot|-||            
              \       \2//

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of (log(cot(x/2)+1))

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Piecewise:

The solution

Method #1

Method #2