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Similar expressions
1/(exp(1/x)+1)

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

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

The solution

You have entered [src]

  1   
------
 1    
 -    
 x    
e  - 1

$$\frac{1}{e^{\frac{1}{x}} - 1}$$

1/(exp(1/x) - 1)

Detail solution

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

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

$\frac{1}{4 x^{2} \sinh^{2}{\left(\frac{1}{2 x} \right)}}$

The answer is:

$\frac{1}{4 x^{2} \sinh^{2}{\left(\frac{1}{2 x} \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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

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The solution