Derivative of log(log(x))

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log(log(x))

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

log(log(x))

Detail solution

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

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph