Mister Exam

Derivative of ln(tg(x/2))

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

The graph:

from to

Piecewise:

The solution

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

  2. The derivative of is .

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