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Simplificación de expresiones/ Descomposición en fracciones simples/ exp(x)/(e^x+2)-(e^x+4)*exp(x)/(e^x+2)^2

¿Cómo vas a descomponer esta exp(x)/(e^x+2)-(e^x+4)*exp(x)/(e^x+2)^2 expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
   x     / x    \  x
  e      \E  + 4/*e 
------ - -----------
 x                2 
E  + 2    / x    \  
          \E  + 2/
−(ex+4)ex(ex+2)2+exex+2- \frac{\left(e^{x} + 4\right) e^{x}}{\left(e^{x} + 2\right)^{2}} + \frac{e^{x}}{e^{x} + 2} 
exp(x)/(E^x + 2) - (E^x + 4)*exp(x)/(E^x + 2)^2
Simplificación general [src] 
      x  
  -2*e   
---------
        2
/     x\ 
\2 + e /
−2ex(ex+2)2- \frac{2 e^{x}}{\left(e^{x} + 2\right)^{2}} 
-2*exp(x)/(2 + exp(x))^2
Descomposición de una fracción [src] 
-2/(2 + exp(x)) + 4/(2 + exp(x))^2
−2ex+2+4(ex+2)2- \frac{2}{e^{x} + 2} + \frac{4}{\left(e^{x} + 2\right)^{2}} 
    2          4    
- ------ + ---------
       x           2
  2 + e    /     x\ 
           \2 + e /
Respuesta numérica [src] 
exp(x)/(2.0 + 2.71828182845905^x) - 0.25*(4.0 + 2.71828182845905^x)*exp(x)/(1 + 0.5*2.71828182845905^x)^2
exp(x)/(2.0 + 2.71828182845905^x) - 0.25*(4.0 + 2.71828182845905^x)*exp(x)/(1 + 0.5*2.71828182845905^x)^2
Unión de expresiones racionales [src] 
      x  
  -2*e   
---------
        2
/     x\ 
\2 + e /
−2ex(ex+2)2- \frac{2 e^{x}}{\left(e^{x} + 2\right)^{2}} 
-2*exp(x)/(2 + exp(x))^2
Potencias [src] 
   x     /      x\  x
  e      \-4 - e /*e 
------ + ------------
     x            2  
2 + e     /     x\   
          \2 + e /
(−ex−4)ex(ex+2)2+exex+2\frac{\left(- e^{x} - 4\right) e^{x}}{\left(e^{x} + 2\right)^{2}} + \frac{e^{x}}{e^{x} + 2} 
   x     /     x\  x
  e      \4 + e /*e 
------ - -----------
     x            2 
2 + e     /     x\  
          \2 + e /
exex+2−(ex+4)ex(ex+2)2\frac{e^{x}}{e^{x} + 2} - \frac{\left(e^{x} + 4\right) e^{x}}{\left(e^{x} + 2\right)^{2}} 
exp(x)/(2 + exp(x)) - (4 + exp(x))*exp(x)/(2 + exp(x))^2
Denominador racional [src] 
        2                          
/     x\   x   /     x\ /     x\  x
\2 + e / *e  - \2 + e /*\4 + e /*e 
-----------------------------------
                     3             
             /     x\              
             \2 + e /
(ex+2)2ex−(ex+2)(ex+4)ex(ex+2)3\frac{\left(e^{x} + 2\right)^{2} e^{x} - \left(e^{x} + 2\right) \left(e^{x} + 4\right) e^{x}}{\left(e^{x} + 2\right)^{3}} 
((2 + exp(x))^2*exp(x) - (2 + exp(x))*(4 + exp(x))*exp(x))/(2 + exp(x))^3
Denominador común [src] 
         x     
     -2*e      
---------------
       x    2*x
4 + 4*e  + e
−2exe2x+4ex+4- \frac{2 e^{x}}{e^{2 x} + 4 e^{x} + 4} 
-2*exp(x)/(4 + 4*exp(x) + exp(2*x))
Compilar la expresión [src] 
   x     /     x\  x
  e      \4 + e /*e 
------ - -----------
     x            2 
2 + e     /     x\  
          \2 + e /
exex+2−(ex+4)ex(ex+2)2\frac{e^{x}}{e^{x} + 2} - \frac{\left(e^{x} + 4\right) e^{x}}{\left(e^{x} + 2\right)^{2}} 
exp(x)/(2 + exp(x)) - (4 + exp(x))*exp(x)/(2 + exp(x))^2
Combinatoria [src] 
      x  
  -2*e   
---------
        2
/     x\ 
\2 + e /
−2ex(ex+2)2- \frac{2 e^{x}}{\left(e^{x} + 2\right)^{2}} 
-2*exp(x)/(2 + exp(x))^2
Parte trigonométrica [src] 
  cosh(x) + sinh(x)     (cosh(x) + sinh(x))*(4 + cosh(x) + sinh(x))
--------------------- - -------------------------------------------
2 + cosh(x) + sinh(x)                                    2         
                                  (2 + cosh(x) + sinh(x))
sinh⁡(x)+cosh⁡(x)sinh⁡(x)+cosh⁡(x)+2−(sinh⁡(x)+cosh⁡(x))(sinh⁡(x)+cosh⁡(x)+4)(sinh⁡(x)+cosh⁡(x)+2)2\frac{\sinh{\left(x \right)} + \cosh{\left(x \right)}}{\sinh{\left(x \right)} + \cosh{\left(x \right)} + 2} - \frac{\left(\sinh{\left(x \right)} + \cosh{\left(x \right)}\right) \left(\sinh{\left(x \right)} + \cosh{\left(x \right)} + 4\right)}{\left(\sinh{\left(x \right)} + \cosh{\left(x \right)} + 2\right)^{2}} 
   -2*(cosh(x) + sinh(x))  
---------------------------
                          2
/                       x\ 
\2 + (cosh(1) + sinh(1)) /
−2(sinh⁡(x)+cosh⁡(x))((sinh⁡(1)+cosh⁡(1))x+2)2- \frac{2 \left(\sinh{\left(x \right)} + \cosh{\left(x \right)}\right)}{\left(\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{x} + 2\right)^{2}} 
                           /                       x\                    
   cosh(x) + sinh(x)       \4 + (cosh(1) + sinh(1)) /*(cosh(x) + sinh(x))
------------------------ - ----------------------------------------------
                       x                                      2          
2 + (cosh(1) + sinh(1))             /                       x\           
                                    \2 + (cosh(1) + sinh(1)) /
sinh⁡(x)+cosh⁡(x)(sinh⁡(1)+cosh⁡(1))x+2−((sinh⁡(1)+cosh⁡(1))x+4)(sinh⁡(x)+cosh⁡(x))((sinh⁡(1)+cosh⁡(1))x+2)2\frac{\sinh{\left(x \right)} + \cosh{\left(x \right)}}{\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{x} + 2} - \frac{\left(\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{x} + 4\right) \left(\sinh{\left(x \right)} + \cosh{\left(x \right)}\right)}{\left(\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{x} + 2\right)^{2}} 
 -2*(cosh(x) + sinh(x)) 
------------------------
                       2
(2 + cosh(x) + sinh(x))
−2(sinh⁡(x)+cosh⁡(x))(sinh⁡(x)+cosh⁡(x)+2)2- \frac{2 \left(\sinh{\left(x \right)} + \cosh{\left(x \right)}\right)}{\left(\sinh{\left(x \right)} + \cosh{\left(x \right)} + 2\right)^{2}} 
   x     /     x\  x
  e      \4 + e /*e 
------ - -----------
     x            2 
2 + e     /     x\  
          \2 + e /
exex+2−(ex+4)ex(ex+2)2\frac{e^{x}}{e^{x} + 2} - \frac{\left(e^{x} + 4\right) e^{x}}{\left(e^{x} + 2\right)^{2}} 
exp(x)/(2 + exp(x)) - (4 + exp(x))*exp(x)/(2 + exp(x))^2
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