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¿Cómo vas a descomponer esta exp(-x)/(1+E^(-x))^2 expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src]
    -x    
   e      
----------
         2
/     -x\ 
\1 + E  / 
$$\frac{e^{- x}}{\left(1 + e^{- x}\right)^{2}}$$
exp(-x)/(1 + E^(-x))^2
Descomposición de una fracción [src]
1/(1 + exp(x)) - 1/(1 + exp(x))^2
$$\frac{1}{e^{x} + 1} - \frac{1}{\left(e^{x} + 1\right)^{2}}$$
  1          1    
------ - ---------
     x           2
1 + e    /     x\ 
         \1 + e / 
Simplificación general [src]
    1     
----------
      2/x\
4*cosh |-|
       \2/
$$\frac{1}{4 \cosh^{2}{\left(\frac{x}{2} \right)}}$$
1/(4*cosh(x/2)^2)
Denominador racional [src]
     x   
    e    
---------
        2
/     x\ 
\1 + e / 
$$\frac{e^{x}}{\left(e^{x} + 1\right)^{2}}$$
exp(x)/(1 + exp(x))^2
Denominador común [src]
        x      
       e       
---------------
       x    2*x
1 + 2*e  + e   
$$\frac{e^{x}}{e^{2 x} + 2 e^{x} + 1}$$
exp(x)/(1 + 2*exp(x) + exp(2*x))
Respuesta numérica [src]
exp(-x)/(1.0 + 2.71828182845905^(-x))^2
exp(-x)/(1.0 + 2.71828182845905^(-x))^2
Unión de expresiones racionales [src]
     x   
    e    
---------
        2
/     x\ 
\1 + e / 
$$\frac{e^{x}}{\left(e^{x} + 1\right)^{2}}$$
exp(x)/(1 + exp(x))^2
Parte trigonométrica [src]
 -(-cosh(x) + sinh(x))  
------------------------
                       2
(1 - sinh(x) + cosh(x)) 
$$- \frac{\sinh{\left(x \right)} - \cosh{\left(x \right)}}{\left(- \sinh{\left(x \right)} + \cosh{\left(x \right)} + 1\right)^{2}}$$
   -(-cosh(x) + sinh(x))    
----------------------------
                           2
/                       -x\ 
\1 + (cosh(1) + sinh(1))  / 
$$- \frac{\sinh{\left(x \right)} - \cosh{\left(x \right)}}{\left(1 + \left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{- x}\right)^{2}}$$
   -sinh(x) + cosh(x)   
------------------------
                       2
(1 - sinh(x) + cosh(x)) 
$$\frac{- \sinh{\left(x \right)} + \cosh{\left(x \right)}}{\left(- \sinh{\left(x \right)} + \cosh{\left(x \right)} + 1\right)^{2}}$$
     -sinh(x) + cosh(x)     
----------------------------
                           2
/                       -x\ 
\1 + (cosh(1) + sinh(1))  / 
$$\frac{- \sinh{\left(x \right)} + \cosh{\left(x \right)}}{\left(1 + \left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{- x}\right)^{2}}$$
(-sinh(x) + cosh(x))/(1 + (cosh(1) + sinh(1))^(-x))^2
Combinatoria [src]
     x   
    e    
---------
        2
/     x\ 
\1 + e / 
$$\frac{e^{x}}{\left(e^{x} + 1\right)^{2}}$$
exp(x)/(1 + exp(x))^2