Derivative of ln(ln(lnx))

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log(log(log(x)))

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

log(log(log(x)))

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of ln(ln(lnx))

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The solution