Mister Exam

Derivative of y=log(log(logx))

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
log(log(log(x)))
log(log(log(x)))\log{\left(\log{\left(\log{\left(x \right)} \right)} \right)}
log(log(log(x)))
Detail solution
  1. Let u=log(log(x))u = \log{\left(\log{\left(x \right)} \right)}.

  2. The derivative of log(u)\log{\left(u \right)} is 1u\frac{1}{u}.

  3. Then, apply the chain rule. Multiply by ddxlog(log(x))\frac{d}{d x} \log{\left(\log{\left(x \right)} \right)}:

    1. Let u=log(x)u = \log{\left(x \right)}.

    2. The derivative of log(u)\log{\left(u \right)} is 1u\frac{1}{u}.

    3. Then, apply the chain rule. Multiply by ddxlog(x)\frac{d}{d x} \log{\left(x \right)}:

      1. The derivative of log(x)\log{\left(x \right)} is 1x\frac{1}{x}.

      The result of the chain rule is:

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

    The result of the chain rule is:

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


The answer is:

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

The graph
02468-8-6-4-2-1010200-100
The first derivative [src]
         1          
--------------------
x*log(x)*log(log(x))
1xlog(x)log(log(x))\frac{1}{x \log{\left(x \right)} \log{\left(\log{\left(x \right)} \right)}}
The second derivative [src]
 /      1              1         \ 
-|1 + ------ + ------------------| 
 \    log(x)   log(x)*log(log(x))/ 
-----------------------------------
        2                          
       x *log(x)*log(log(x))       
1+1log(x)+1log(x)log(log(x))x2log(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)}}
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))                                 
2+3log(x)+3log(x)log(log(x))+2log(x)2+3log(x)2log(log(x))+2log(x)2log(log(x))2x3log(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)}}
The graph
Derivative of y=log(log(logx))