Derivative of log(log(x)+1/x)

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

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

log(log(x) + 1/x)

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of log(log(x)+1/x)

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The solution