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Graphing y =:
1/(1+e^(-x))
Integral of d{x}:
1/(1+e^(-x))
Limit of the function:
1/(1+e^(-x))
Identical expressions
one /(one +e^(-x))
1 divide by (1 plus e to the power of ( minus x))
one divide by (one plus e to the power of ( minus x))
1/(1+e(-x))
1/1+e-x
1/1+e^-x
1 divide by (1+e^(-x))
Similar expressions
1/(1-e^(-x))
1/(1+e^-x)+1/(1+e^x)
1/(1+e^(x))

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

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

The solution

You have entered [src]

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

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

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

Detail solution

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

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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

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The solution