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Derivative of:
Derivative of 5x
Derivative of 5x^3
Derivative of 2*cos(x)
Derivative of x*cos(2*x)
Identical expressions
sqrt(tg^ three (x/ two))
square root of (tg cubed (x divide by 2))
square root of (tg to the power of three (x divide by two))
√(tg^3(x/2))
sqrt(tg3(x/2))
sqrttg3x/2
sqrt(tg³(x/2))
sqrt(tg to the power of 3(x/2))
sqrttg^3x/2
sqrt(tg^3(x divide by 2))

The derivative of the function/ sqrt(tg^3(x/2))

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

The solution

You have entered [src]

    _________
   /    3/x\ 
  /  tan |-| 
\/       \2/

$$\sqrt{\tan^{3}{\left(\frac{x}{2} \right)}}$$

Transform

Detail solution

Let $u = \tan^{3}{\left(\frac{x}{2} \right)}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan^{3}{\left(\frac{x}{2} \right)}$ :
1. Let $u = \tan{\left(\frac{x}{2} \right)}$ .
2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
3. 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.
        
        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.
        
        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:
  
  $\frac{3 \left(\frac{\sin^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{2}\right) \tan^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}$
The result of the chain rule is:

$\frac{3 \left(\frac{\sin^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{2}\right) \tan^{2}{\left(\frac{x}{2} \right)}}{2 \sqrt{\tan^{3}{\left(\frac{x}{2} \right)}} \cos^{2}{\left(\frac{x}{2} \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

              /         2/x\\
    _________ |    3*tan |-||
   /    3/x\  |3         \2/|
  /  tan |-| *|- + ---------|
\/       \2/  \2       2    /
-----------------------------
                /x\          
           2*tan|-|          
                \2/

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

Simplify

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

$$\frac{3 \left(\frac{\tan^{2}{\left(\frac{x}{2} \right)} + 1}{\tan^{2}{\left(\frac{x}{2} \right)}} + 4\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \sqrt{\tan^{3}{\left(\frac{x}{2} \right)}}}{16}$$

Simplify

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

$$\frac{3 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(- \frac{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{\tan^{3}{\left(\frac{x}{2} \right)}} + \frac{20 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}{\tan{\left(\frac{x}{2} \right)}} + 16 \tan{\left(\frac{x}{2} \right)}\right) \sqrt{\tan^{3}{\left(\frac{x}{2} \right)}}}{64}$$

Simplify

The graph

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Identical expressions

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

The graph:

Piecewise:

The solution