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

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   2                 
log (x) - log(log(x))

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

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

Detail solution

Differentiate $\log{\left(x \right)}^{2} - \log{\left(\log{\left(x \right)} \right)}$ term by term:
1. Let $u = \log{\left(x \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
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:
  
  $\frac{2 \log{\left(x \right)}}{x}$
4. The derivative of a constant times a function is the constant times the derivative of the function.
  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:
    
    $\frac{1}{x \log{\left(x \right)}}$
  So, the result is: $- \frac{1}{x \log{\left(x \right)}}$
The result is: $\frac{2 \log{\left(x \right)}}{x} - \frac{1}{x \log{\left(x \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

        3        2         2              
-6 - ------- - ------ - ------- + 4*log(x)
        2      log(x)      3              
     log (x)            log (x)           
------------------------------------------
                     3                    
                    x

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

Simplify

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

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The solution