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The derivative of the function/ log(log(x)+1/x)

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

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

The graph:

from to

Piecewise:

The solution

You have entered [src] 
   /         1\
log|log(x) + -|
   \         x/
log⁡(log⁡(x)+1x)\log{\left(\log{\left(x \right)} + \frac{1}{x} \right)} 
log(log(x) + 1/x)
Detail solution

  1. Let u=log⁡(x)+1xu = \log{\left(x \right)} + \frac{1}{x}.

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

  3. Then, apply the chain rule. Multiply by ddx(log⁡(x)+1x)\frac{d}{d x} \left(\log{\left(x \right)} + \frac{1}{x}\right):

    1. Differentiate log⁡(x)+1x\log{\left(x \right)} + \frac{1}{x} term by term:

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

      2. Apply the power rule: 1x\frac{1}{x} goes to −1x2- \frac{1}{x^{2}}

      The result is: 1x−1x2\frac{1}{x} - \frac{1}{x^{2}}

    The result of the chain rule is:

    1x−1x2log⁡(x)+1x\frac{\frac{1}{x} - \frac{1}{x^{2}}}{\log{\left(x \right)} + \frac{1}{x}}

  4. Now simplify:

    x−1x(xlog⁡(x)+1)\frac{x - 1}{x \left(x \log{\left(x \right)} + 1\right)}

The answer is:

x−1x(xlog⁡(x)+1)\frac{x - 1}{x \left(x \log{\left(x \right)} + 1\right)}

The graph
02468-8-6-4-2-1010-2010
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
  1   1   
  - - --  
  x    2  
      x   
----------
         1
log(x) + -
         x
1x−1x2log⁡(x)+1x\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         /
−(1−1x)2log⁡(x)+1x+1−2xx2(log⁡(x)+1x)- \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         /
3(1−2x)(1−1x)log⁡(x)+1x+2(1−1x)3(log⁡(x)+1x)2+2−6xx3(log⁡(x)+1x)\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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