Derivative of log2[log3(log2x)]

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   /log(log(2*x))\
log|-------------|
   \    log(3)   /
------------------
      log(2)

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

  /   /log(log(2*x))\\
  |log|-------------||
d |   \    log(3)   /|
--|------------------|
dx\      log(2)      /

$$\frac{d}{d x} \frac{\log{\left(\frac{\log{\left(\log{\left(2 x \right)} \right)}}{\log{\left(3 \right)}} \right)}}{\log{\left(2 \right)}}$$

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Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \frac{\log{\left(\log{\left(2 x \right)} \right)}}{\log{\left(3 \right)}}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{\log{\left(\log{\left(2 x \right)} \right)}}{\log{\left(3 \right)}}$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Let $u = \log{\left(2 x \right)}$ .
    2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(2 x \right)}$ :
      1. Let $u = 2 x$ .
      2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
        
        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: $2$
        
        The result of the chain rule is:
        
        $\frac{1}{x}$
      The result of the chain rule is:
      
      $\frac{1}{x \log{\left(2 x \right)}}$
    So, the result is: $\frac{1}{x \log{\left(3 \right)} \log{\left(2 x \right)}}$
  The result of the chain rule is:
  
  $\frac{1}{x \log{\left(2 x \right)} \log{\left(\log{\left(2 x \right)} \right)}}$
So, the result is: $\frac{1}{x \log{\left(2 \right)} \log{\left(2 x \right)} \log{\left(\log{\left(2 x \right)} \right)}}$

The answer is:

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

The graph

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Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

        2          3                  2                         3                         3           
2 + --------- + -------- + ------------------------ + ---------------------- + -----------------------
       2        log(2*x)      2         2             log(2*x)*log(log(2*x))      2                   
    log (2*x)              log (2*x)*log (log(2*x))                            log (2*x)*log(log(2*x))
------------------------------------------------------------------------------------------------------
                                    3                                                                 
                                   x *log(2)*log(2*x)*log(log(2*x))

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

Simplify

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Derivative of log2[log3(log2x)]

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Piecewise:

The solution