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sqrt(tg^3(x/2))

Derivative of sqrt(tg^3(x/2))

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
    _________
   /    3/x\ 
  /  tan |-| 
\/       \2/ 
$$\sqrt{\tan^{3}{\left(\frac{x}{2} \right)}}$$
Detail solution
  1. Let .

  2. Apply the power rule: goes to

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

    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:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

The graph
The first derivative [src]
              /         2/x\\
    _________ |    3*tan |-||
   /    3/x\  |3         \2/|
  /  tan |-| *|- + ---------|
\/       \2/  \2       2    /
-----------------------------
                /x\          
           2*tan|-|          
                \2/          
$$\frac{\left(\frac{3 \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{3}{2}\right) \sqrt{\tan^{3}{\left(\frac{x}{2} \right)}}}{2 \tan{\left(\frac{x}{2} \right)}}$$
The second derivative [src]
                              /           2/x\\
      _________               |    1 + tan |-||
     /    3/x\  /       2/x\\ |            \2/|
3*  /  tan |-| *|1 + tan |-||*|4 + -----------|
  \/       \2/  \        \2// |         2/x\  |
                              |      tan |-|  |
                              \          \2/  /
-----------------------------------------------
                       16                      
$$\frac{3 \left(\frac{\tan^{2}{\left(\frac{x}{2} \right)} + 1}{\tan^{2}{\left(\frac{x}{2} \right)}} + 4\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \sqrt{\tan^{3}{\left(\frac{x}{2} \right)}}}{16}$$
The third derivative [src]
                              /                         2                   \
                              |            /       2/x\\       /       2/x\\|
      _________               |            |1 + tan |-||    20*|1 + tan |-|||
     /    3/x\  /       2/x\\ |      /x\   \        \2//       \        \2//|
3*  /  tan |-| *|1 + tan |-||*|16*tan|-| - -------------- + ----------------|
  \/       \2/  \        \2// |      \2/         3/x\               /x\     |
                              |               tan |-|            tan|-|     |
                              \                   \2/               \2/     /
-----------------------------------------------------------------------------
                                      64                                     
$$\frac{3 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(- \frac{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{\tan^{3}{\left(\frac{x}{2} \right)}} + \frac{20 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}{\tan{\left(\frac{x}{2} \right)}} + 16 \tan{\left(\frac{x}{2} \right)}\right) \sqrt{\tan^{3}{\left(\frac{x}{2} \right)}}}{64}$$
The graph
Derivative of sqrt(tg^3(x/2))