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Identical expressions
lntg^ two *(x/ six)
lntg squared multiply by (x divide by 6)
lntg to the power of two multiply by (x divide by six)
lntg2*(x/6)
lntg2*x/6
lntg²*(x/6)
lntg to the power of 2*(x/6)
lntg^2(x/6)
lntg2(x/6)
lntg2x/6
lntg^2x/6
lntg^2*(x divide by 6)
Similar expressions
ln*tg^2(x/6)

The derivative of the function/ lntg^2*(x/6)

Derivative of lntg^2*(x/6)

The solution

You have entered [src]

   2/   /x\\
log |tan|-||
    \   \6//

\log{\left(\tan{\left(\frac{x}{6} \right)} \right)}^{2}

d /   2/   /x\\\
--|log |tan|-|||
dx\    \   \6///

\frac{d}{d x} \log{\left(\tan{\left(\frac{x}{6} \right)} \right)}^{2}

Transform

Detail solution

Let $u = \log{\left(\tan{\left(\frac{x}{6} \right)} \right)}$ .
Apply the power rule: $u^{2}$ goes to $2 u$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(\tan{\left(\frac{x}{6} \right)} \right)}$ :
1. Let $u = \tan{\left(\frac{x}{6} \right)}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(\frac{x}{6} \right)}$ :
  1. Rewrite the function to be differentiated:
    
    $\tan{\left(\frac{x}{6} \right)} = \frac{\sin{\left(\frac{x}{6} \right)}}{\cos{\left(\frac{x}{6} \right)}}$
  2. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \sin{\left(\frac{x}{6} \right)}$ and $g{\left(x \right)} = \cos{\left(\frac{x}{6} \right)}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Let $u = \frac{x}{6}$ .
    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 $\frac{d}{d x} \frac{x}{6}$ :
      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: $\frac{1}{6}$
      The result of the chain rule is:
      
      $\frac{\cos{\left(\frac{x}{6} \right)}}{6}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Let $u = \frac{x}{6}$ .
    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 $\frac{d}{d x} \frac{x}{6}$ :
      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: $\frac{1}{6}$
      The result of the chain rule is:
      
      $- \frac{\sin{\left(\frac{x}{6} \right)}}{6}$
    Now plug in to the quotient rule:
    
    $\frac{\frac{\sin^{2}{\left(\frac{x}{6} \right)}}{6} + \frac{\cos^{2}{\left(\frac{x}{6} \right)}}{6}}{\cos^{2}{\left(\frac{x}{6} \right)}}$
  The result of the chain rule is:
  
  $\frac{\frac{\sin^{2}{\left(\frac{x}{6} \right)}}{6} + \frac{\cos^{2}{\left(\frac{x}{6} \right)}}{6}}{\cos^{2}{\left(\frac{x}{6} \right)} \tan{\left(\frac{x}{6} \right)}}$
The result of the chain rule is:

$\frac{2 \left(\frac{\sin^{2}{\left(\frac{x}{6} \right)}}{6} + \frac{\cos^{2}{\left(\frac{x}{6} \right)}}{6}\right) \log{\left(\tan{\left(\frac{x}{6} \right)} \right)}}{\cos^{2}{\left(\frac{x}{6} \right)} \tan{\left(\frac{x}{6} \right)}}$
Now simplify:

$\frac{2 \log{\left(\tan{\left(\frac{x}{6} \right)} \right)}}{3 \sin{\left(\frac{x}{3} \right)}}$

The answer is:

\frac{2 \log{\left(\tan{\left(\frac{x}{6} \right)} \right)}}{3 \sin{\left(\frac{x}{3} \right)}}

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  /       2/x\\            
  |    tan |-||            
  |1       \6/|    /   /x\\
2*|- + -------|*log|tan|-||
  \6      6   /    \   \6//
---------------------------
              /x\          
           tan|-|          
              \6/

\frac{2 \left(\frac{\tan^{2}{\left(\frac{x}{6} \right)}}{6} + \frac{1}{6}\right) \log{\left(\tan{\left(\frac{x}{6} \right)} \right)}}{\tan{\left(\frac{x}{6} \right)}}

Simplify

The second derivative [src]

              /                       2/x\   /       2/x\\    /   /x\\\
              |                1 + tan |-|   |1 + tan |-||*log|tan|-|||
/       2/x\\ |     /   /x\\           \6/   \        \6//    \   \6//|
|1 + tan |-||*|2*log|tan|-|| + ----------- - -------------------------|
\        \6// |     \   \6//        2/x\                 2/x\         |
              |                  tan |-|              tan |-|         |
              \                      \6/                  \6/         /
-----------------------------------------------------------------------
                                   18

\frac{\left(\tan^{2}{\left(\frac{x}{6} \right)} + 1\right) \left(- \frac{\left(\tan^{2}{\left(\frac{x}{6} \right)} + 1\right) \log{\left(\tan{\left(\frac{x}{6} \right)} \right)}}{\tan^{2}{\left(\frac{x}{6} \right)}} + \frac{\tan^{2}{\left(\frac{x}{6} \right)} + 1}{\tan^{2}{\left(\frac{x}{6} \right)}} + 2 \log{\left(\tan{\left(\frac{x}{6} \right)} \right)}\right)}{18}

Simplify

The third derivative [src]

              /                 2                                                                                         2            \
              |    /       2/x\\                             /       2/x\\     /       2/x\\    /   /x\\     /       2/x\\     /   /x\\|
              |  3*|1 + tan |-||                           6*|1 + tan |-||   4*|1 + tan |-||*log|tan|-||   2*|1 + tan |-|| *log|tan|-|||
/       2/x\\ |    \        \6//         /   /x\\    /x\     \        \6//     \        \6//    \   \6//     \        \6//     \   \6//|
|1 + tan |-||*|- ---------------- + 4*log|tan|-||*tan|-| + --------------- - --------------------------- + ----------------------------|
\        \6// |         3/x\             \   \6//    \6/           /x\                     /x\                          3/x\           |
              |      tan |-|                                    tan|-|                  tan|-|                       tan |-|           |
              \          \6/                                       \6/                     \6/                           \6/           /
----------------------------------------------------------------------------------------------------------------------------------------
                                                                  108

\frac{\left(\tan^{2}{\left(\frac{x}{6} \right)} + 1\right) \left(\frac{2 \left(\tan^{2}{\left(\frac{x}{6} \right)} + 1\right)^{2} \log{\left(\tan{\left(\frac{x}{6} \right)} \right)}}{\tan^{3}{\left(\frac{x}{6} \right)}} - \frac{3 \left(\tan^{2}{\left(\frac{x}{6} \right)} + 1\right)^{2}}{\tan^{3}{\left(\frac{x}{6} \right)}} - \frac{4 \left(\tan^{2}{\left(\frac{x}{6} \right)} + 1\right) \log{\left(\tan{\left(\frac{x}{6} \right)} \right)}}{\tan{\left(\frac{x}{6} \right)}} + \frac{6 \left(\tan^{2}{\left(\frac{x}{6} \right)} + 1\right)}{\tan{\left(\frac{x}{6} \right)}} + 4 \log{\left(\tan{\left(\frac{x}{6} \right)} \right)} \tan{\left(\frac{x}{6} \right)}\right)}{108}

Simplify

The graph

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Derivative of lntg^2*(x/6)

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The solution