Derivative of ln(tgx/2)

The solution

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

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

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

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

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Detail solution

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

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

$\frac{2}{\sin{\left(2 x \right)}}$

The answer is:

$\frac{2}{\sin{\left(2 x \right)}}$

The graph

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The first derivative [src]

  /       2   \
  |1   tan (x)|
2*|- + -------|
  \2      2   /
---------------
     tan(x)

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

Simplify

The second derivative [src]

                             2
                /       2   \ 
         2      \1 + tan (x)/ 
2 + 2*tan (x) - --------------
                      2       
                   tan (x)

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of ln(tgx/2)

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