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Identical expressions
(one /(x))/(exp^(one /x)- one)
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(one divide by (x)) divide by ( exponent of to the power of (one divide by x) minus one)
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Similar expressions
(1/(x))/(exp^(1/x)+1)

The derivative of the function/ (1/(x))/(exp^(1/x)-1)

Derivative of (1/(x))/(exp^(1/x)-1)

The solution

You have entered [src]

      1      
-------------
  /x ___    \
x*\\/ E  - 1/

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

1/(x*(E^(1/x) - 1))

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = 1$ and $g{\left(x \right)} = x \left(e^{\frac{1}{x}} - 1\right)$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of the constant $1$ is zero.
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  $g{\left(x \right)} = e^{\frac{1}{x}} - 1$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Differentiate $e^{\frac{1}{x}} - 1$ term by term:
    1. The derivative of the constant $-1$ is zero.
    2. Let $u = \frac{1}{x}$ .
    3. The derivative of $e^{u}$ is itself.
    4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{1}{x}$ :
      1. Apply the power rule: $\frac{1}{x}$ goes to $- \frac{1}{x^{2}}$
      The result of the chain rule is:
      
      $- \frac{e^{\frac{1}{x}}}{x^{2}}$
    The result is: $- \frac{e^{\frac{1}{x}}}{x^{2}}$
  The result is: $e^{\frac{1}{x}} - 1 - \frac{e^{\frac{1}{x}}}{x}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                           1      
                           -      
                           x      
        1                 e       
- -------------- + ---------------
   2 /x ___    \                 2
  x *\\/ E  - 1/    3 /x ___    \ 
                   x *\\/ E  - 1/

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

Simplify

The second derivative [src]

                  /               1   \   
                  |               -   |  1
                  |               x   |  -
                  |    1       2*e    |  x
                  |2 + - - -----------|*e 
           1      |    x     /      1\|   
           -      |          |      -||   
           x      |          |      x||   
        2*e       \        x*\-1 + e //   
2 - ----------- - ------------------------
      /      1\           /      1\       
      |      -|           |      -|       
      |      x|           |      x|       
    x*\-1 + e /         x*\-1 + e /       
------------------------------------------
                  /      1\               
                  |      -|               
                3 |      x|               
               x *\-1 + e /

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

Simplify

The third derivative [src]

                   /                    1             1               2    \                                
                   |                    -             -               -    |  1     /               1   \   
                   |                    x             x               x    |  -     |               -   |  1
                   |    1    6      12*e           6*e             6*e     |  x     |               x   |  -
                   |6 + -- + - - ----------- - ------------ + -------------|*e      |    1       2*e    |  x
                   |     2   x     /      1\      /      1\               2|      3*|2 + - - -----------|*e 
            1      |    x          |      -|      |      -|      /      1\ |        |    x     /      1\|   
            -      |               |      x|    2 |      x|      |      -| |        |          |      -||   
            x      |             x*\-1 + e /   x *\-1 + e /    2 |      x| |        |          |      x||   
         6*e       \                                          x *\-1 + e / /        \        x*\-1 + e //   
-6 + ----------- + ------------------------------------------------------------ + --------------------------
       /      1\                             /      1\                                     /      1\        
       |      -|                             |      -|                                     |      -|        
       |      x|                             |      x|                                     |      x|        
     x*\-1 + e /                           x*\-1 + e /                                   x*\-1 + e /        
------------------------------------------------------------------------------------------------------------
                                                   /      1\                                                
                                                   |      -|                                                
                                                 4 |      x|                                                
                                                x *\-1 + e /

$$\frac{-6 + \frac{3 \left(2 + \frac{1}{x} - \frac{2 e^{\frac{1}{x}}}{x \left(e^{\frac{1}{x}} - 1\right)}\right) e^{\frac{1}{x}}}{x \left(e^{\frac{1}{x}} - 1\right)} + \frac{\left(6 + \frac{6}{x} - \frac{12 e^{\frac{1}{x}}}{x \left(e^{\frac{1}{x}} - 1\right)} + \frac{1}{x^{2}} - \frac{6 e^{\frac{1}{x}}}{x^{2} \left(e^{\frac{1}{x}} - 1\right)} + \frac{6 e^{\frac{2}{x}}}{x^{2} \left(e^{\frac{1}{x}} - 1\right)^{2}}\right) e^{\frac{1}{x}}}{x \left(e^{\frac{1}{x}} - 1\right)} + \frac{6 e^{\frac{1}{x}}}{x \left(e^{\frac{1}{x}} - 1\right)}}{x^{4} \left(e^{\frac{1}{x}} - 1\right)}$$

Simplify

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Derivative of (1/(x))/(exp^(1/x)-1)

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The solution