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

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

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

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The solution

You have entered [src] 
   x  
  E   
------
     x
1 + E
exex+1\frac{e^{x}}{e^{x} + 1} 
E^x/(1 + E^x)
Detail solution

  1. Apply the quotient rule, which is:

    ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=exf{\left(x \right)} = e^{x} and g(x)=ex+1g{\left(x \right)} = e^{x} + 1.

    To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. The derivative of exe^{x} is itself.

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Differentiate ex+1e^{x} + 1 term by term:

      1. The derivative of the constant 11 is zero.

      2. The derivative of exe^{x} is itself.

      The result is: exe^{x}

    Now plug in to the quotient rule:

    (ex+1)ex−e2x(ex+1)2\frac{\left(e^{x} + 1\right) e^{x} - e^{2 x}}{\left(e^{x} + 1\right)^{2}}

  2. Now simplify:

    14cosh⁡2(x2)\frac{1}{4 \cosh^{2}{\left(\frac{x}{2} \right)}}

The answer is:

14cosh⁡2(x2)\frac{1}{4 \cosh^{2}{\left(\frac{x}{2} \right)}}

The graph
02468-8-6-4-2-101002
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Plot the graph f'(x)
The first derivative [src] 
   x         2*x  
  e         e     
------ - ---------
     x           2
1 + E    /     x\ 
         \1 + E /
exex+1−e2x(ex+1)2\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
(−(1−2exex+1)exex+1+1−2exex+1)exex+1\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
(−3(1−2exex+1)exex+1+1−(1−6exex+1+6e2x(ex+1)2)exex+1−3exex+1)exex+1\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
Derivative of e^x/(1+e^x)
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