Derivative of log[tan(x/2)]

The solution

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   /   /x\\
log|tan|-||
   \   \2//

$$\log{\left(\tan{\left(\frac{x}{2} \right)} \right)}$$

d /   /   /x\\\
--|log|tan|-|||
dx\   \   \2///

$$\frac{d}{d x} \log{\left(\tan{\left(\frac{x}{2} \right)} \right)}$$

Transform

Detail solution

Let $u = \tan{\left(\frac{x}{2} \right)}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(\frac{x}{2} \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\tan{\left(\frac{x}{2} \right)} = \frac{\sin{\left(\frac{x}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(\frac{x}{2} \right)}$ and $g{\left(x \right)} = \cos{\left(\frac{x}{2} \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = \frac{x}{2}$ .
  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 $\frac{d}{d x} \frac{x}{2}$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. 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 $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = \frac{x}{2}$ .
  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 $\frac{d}{d x} \frac{x}{2}$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. 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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

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

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

Simplify

The graph

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Derivative of log[tan(x/2)]

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The solution