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tan^- one (x/ two)
tangent of to the power of minus 1(x divide by 2)
tangent of to the power of minus one (x divide by two)
tan-1(x/2)
tan-1x/2
tan^-1x/2
tan^-1(x divide by 2)
Similar expressions
tan^+1(x/2)

The derivative of the function/ tan^-1(x/2)

Derivative of tan^-1(x/2)

The solution

You have entered [src]

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

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

d /  1   \
--|------|
dx|   /x\|
  |tan|-||
  \   \2//

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

Transform

Detail solution

Let $u = \tan{\left(\frac{x}{2} \right)}$ .
Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

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

$$\frac{- \frac{3 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}}{\tan^{4}{\left(\frac{x}{2} \right)}} + \frac{5 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{\tan^{2}{\left(\frac{x}{2} \right)}} - 2 \tan^{2}{\left(\frac{x}{2} \right)} - 2}{4}$$

Simplify

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Derivative of tan^-1(x/2)

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Piecewise:

The solution