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Derivative of:
Derivative of 2x²
Derivative of f(x)=1
Derivative of e^x*x^2
Derivative of 2*sqrt(x^3)
Identical expressions
ln*tg^ two (x/ six)
ln multiply by tg squared (x divide by 6)
ln multiply by tg to the power of two (x divide by six)
ln*tg2(x/6)
ln*tg2x/6
ln*tg²(x/6)
ln*tg to the power of 2(x/6)
lntg^2(x/6)
lntg2(x/6)
lntg2x/6
lntg^2x/6
ln*tg^2(x divide by 6)

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

Derivative of ln*tg^2(x/6)

The solution

You have entered [src]

          2/x\
log(x)*tan |-|
           \6/

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

d /          2/x\\
--|log(x)*tan |-||
dx\           \6//

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

Transform

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = \log{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
$g{\left(x \right)} = \tan^{2}{\left(\frac{x}{6} \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \tan{\left(\frac{x}{6} \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
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:
  
  $\frac{2 \left(\frac{\sin^{2}{\left(\frac{x}{6} \right)}}{6} + \frac{\cos^{2}{\left(\frac{x}{6} \right)}}{6}\right) \tan{\left(\frac{x}{6} \right)}}{\cos^{2}{\left(\frac{x}{6} \right)}}$
The result is: $\frac{2 \left(\frac{\sin^{2}{\left(\frac{x}{6} \right)}}{6} + \frac{\cos^{2}{\left(\frac{x}{6} \right)}}{6}\right) \log{\left(x \right)} \tan{\left(\frac{x}{6} \right)}}{\cos^{2}{\left(\frac{x}{6} \right)}} + \frac{\tan^{2}{\left(\frac{x}{6} \right)}}{x}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   2/x\   /       2/x\\              
tan |-|   |    tan |-||              
    \6/   |1       \6/|           /x\
------- + |- + -------|*log(x)*tan|-|
   x      \3      3   /           \6/

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

Simplify

The second derivative [src]

     2/x\   /       2/x\\ /         2/x\\            /       2/x\\    /x\
  tan |-|   |1 + tan |-||*|1 + 3*tan |-||*log(x)   2*|1 + tan |-||*tan|-|
      \6/   \        \6// \          \6//            \        \6//    \6/
- ------- + ------------------------------------ + ----------------------
      2                      18                             3*x          
     x

$$\frac{\left(\tan^{2}{\left(\frac{x}{6} \right)} + 1\right) \left(3 \tan^{2}{\left(\frac{x}{6} \right)} + 1\right) \log{\left(x \right)}}{18} + \frac{2 \left(\tan^{2}{\left(\frac{x}{6} \right)} + 1\right) \tan{\left(\frac{x}{6} \right)}}{3 x} - \frac{\tan^{2}{\left(\frac{x}{6} \right)}}{x^{2}}$$

Simplify

The third derivative [src]

     2/x\   /       2/x\\    /x\   /       2/x\\ /         2/x\\   /       2/x\\ /         2/x\\           /x\
2*tan |-|   |1 + tan |-||*tan|-|   |1 + tan |-||*|1 + 3*tan |-||   |1 + tan |-||*|2 + 3*tan |-||*log(x)*tan|-|
      \6/   \        \6//    \6/   \        \6// \          \6//   \        \6// \          \6//           \6/
--------- - -------------------- + ----------------------------- + -------------------------------------------
     3                2                         6*x                                     27                    
    x                x

$$\frac{\left(\tan^{2}{\left(\frac{x}{6} \right)} + 1\right) \left(3 \tan^{2}{\left(\frac{x}{6} \right)} + 2\right) \log{\left(x \right)} \tan{\left(\frac{x}{6} \right)}}{27} + \frac{\left(\tan^{2}{\left(\frac{x}{6} \right)} + 1\right) \left(3 \tan^{2}{\left(\frac{x}{6} \right)} + 1\right)}{6 x} - \frac{\left(\tan^{2}{\left(\frac{x}{6} \right)} + 1\right) \tan{\left(\frac{x}{6} \right)}}{x^{2}} + \frac{2 \tan^{2}{\left(\frac{x}{6} \right)}}{x^{3}}$$

Simplify

The graph

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Identical expressions

Derivative of ln*tg^2(x/6)

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The solution