Mister Exam

Derivative of 1/(tg(x/2))

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1   
------
   /x\
tan|-|
   \2/
$$\frac{1}{\tan{\left(\frac{x}{2} \right)}}$$
1/tan(x/2)
Detail solution
  1. Let .

  2. Apply the power rule: goes to

  3. Then, apply the chain rule. Multiply by :

    1. Rewrite the function to be differentiated:

    2. Apply the quotient rule, which is:

      and .

      To find :

      1. Let .

      2. The derivative of sine is cosine:

      3. Then, apply the chain rule. Multiply by :

        1. The derivative of a constant times a function is the constant times the derivative of the function.

          1. Apply the power rule: goes to

          So, the result is:

        The result of the chain rule is:

      To find :

      1. Let .

      2. The derivative of cosine is negative sine:

      3. Then, apply the chain rule. Multiply by :

        1. The derivative of a constant times a function is the constant times the derivative of the function.

          1. Apply the power rule: goes to

          So, the result is:

        The result of the chain rule is:

      Now plug in to the quotient rule:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

The graph
The first derivative [src]
         2/x\
      tan |-|
  1       \2/
- - - -------
  2      2   
-------------
      2/x\   
   tan |-|   
       \2/   
$$\frac{- \frac{\tan^{2}{\left(\frac{x}{2} \right)}}{2} - \frac{1}{2}}{\tan^{2}{\left(\frac{x}{2} \right)}}$$
The second derivative [src]
              /            2/x\\
              |     1 + tan |-||
/       2/x\\ |             \2/|
|1 + tan |-||*|-1 + -----------|
\        \2// |          2/x\  |
              |       tan |-|  |
              \           \2/  /
--------------------------------
                 /x\            
            2*tan|-|            
                 \2/            
$$\frac{\left(\frac{\tan^{2}{\left(\frac{x}{2} \right)} + 1}{\tan^{2}{\left(\frac{x}{2} \right)}} - 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}{2 \tan{\left(\frac{x}{2} \right)}}$$
The third derivative [src]
                                3                  2
                   /       2/x\\      /       2/x\\ 
                 3*|1 + tan |-||    5*|1 + tan |-|| 
          2/x\     \        \2//      \        \2// 
-2 - 2*tan |-| - ---------------- + ----------------
           \2/          4/x\               2/x\     
                     tan |-|            tan |-|     
                         \2/                \2/     
----------------------------------------------------
                         4                          
$$\frac{- \frac{3 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}}{\tan^{4}{\left(\frac{x}{2} \right)}} + \frac{5 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{\tan^{2}{\left(\frac{x}{2} \right)}} - 2 \tan^{2}{\left(\frac{x}{2} \right)} - 2}{4}$$