Derivative of ln(ln(ln(x^2)))

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   /   /   / 2\\\
log\log\log\x ///

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

log(log(log(x^2)))

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

          2           
----------------------
     / 2\    /   / 2\\
x*log\x /*log\log\x //

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Identical expressions

Derivative of ln(ln(ln(x^2)))

The graph:

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The solution