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Similar expressions
e^x/(e^x-1)

The derivative of the function/ e^x/(e^x+1)

Derivative of e^x/(e^x+1)

The solution

You have entered [src]

   x  
  E   
------
 x    
E  + 1

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

E^x/(E^x + 1)

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of $e^{x}$ is itself.
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $e^{x} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. The derivative of $e^{x}$ is itself.
  The result is: $e^{x}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of e^x/(e^x+1)

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Piecewise:

The solution