Derivative of y=log(log(logtanx))

The solution

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log(log(log(tan(x))))

$$\log{\left(\log{\left(\log{\left(\tan{\left(x \right)} \right)} \right)} \right)}$$

d                        
--(log(log(log(tan(x)))))
dx

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

Transform

Detail solution

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

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

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

The answer is:

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

The graph

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Plot the graph f'(x)

The first derivative [src]

                   2               
            1 + tan (x)            
-----------------------------------
log(log(tan(x)))*log(tan(x))*tan(x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of y=log(log(logtanx))

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