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(x*tanx)/(1+e^x)
Derivada de la función/ 1+e^x/ x*tanx/ (x*tanx)/(1+e^x)

Derivada de (x*tanx)/(1+e^x)

Función f() - derivada -er orden en el punto
v

Gráfico:

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Solución

Ha introducido [src] 
x*tan(x)
--------
      x 
 1 + E
xtan⁡(x)ex+1\frac{x \tan{\left(x \right)}}{e^{x} + 1} 
(x*tan(x))/(1 + E^x)
Solución detallada

  1. Se aplica la regla de la derivada parcial:

    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)=xtan⁡(x)f{\left(x \right)} = x \tan{\left(x \right)} y g(x)=ex+1g{\left(x \right)} = e^{x} + 1.

    Para calcular ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Se aplica la regla de la derivada de una multiplicación:

      ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}

      f(x)=xf{\left(x \right)} = x; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. Según el principio, aplicamos: xx tenemos 11

      g(x)=tan⁡(x)g{\left(x \right)} = \tan{\left(x \right)}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. Reescribimos las funciones para diferenciar:

        tan⁡(x)=sin⁡(x)cos⁡(x)\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}

      2. Se aplica la regla de la derivada parcial:

        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)=sin⁡(x)f{\left(x \right)} = \sin{\left(x \right)} y g(x)=cos⁡(x)g{\left(x \right)} = \cos{\left(x \right)}.

        Para calcular ddxf(x)\frac{d}{d x} f{\left(x \right)}:

        1. La derivada del seno es igual al coseno:

          ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

        Para calcular ddxg(x)\frac{d}{d x} g{\left(x \right)}:

        1. La derivada del coseno es igual a menos el seno:

          ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

        Ahora aplicamos la regla de la derivada de una divesión:

        sin⁡2(x)+cos⁡2(x)cos⁡2(x)\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

      Como resultado de: x(sin⁡2(x)+cos⁡2(x))cos⁡2(x)+tan⁡(x)\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + \tan{\left(x \right)}

    Para calcular ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. diferenciamos ex+1e^{x} + 1 miembro por miembro:

      1. La derivada de una constante 11 es igual a cero.

      2. Derivado exe^{x} es.

      Como resultado de: exe^{x}

    Ahora aplicamos la regla de la derivada de una divesión:

    −xextan⁡(x)+(x(sin⁡2(x)+cos⁡2(x))cos⁡2(x)+tan⁡(x))(ex+1)(ex+1)2\frac{- x e^{x} \tan{\left(x \right)} + \left(\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + \tan{\left(x \right)}\right) \left(e^{x} + 1\right)}{\left(e^{x} + 1\right)^{2}}

  2. Simplificamos:

    −xexsin⁡(2x)2+(x+sin⁡(2x)2)(ex+1)(ex+1)2cos⁡2(x)\frac{- \frac{x e^{x} \sin{\left(2 x \right)}}{2} + \left(x + \frac{\sin{\left(2 x \right)}}{2}\right) \left(e^{x} + 1\right)}{\left(e^{x} + 1\right)^{2} \cos^{2}{\left(x \right)}}

Respuesta:

−xexsin⁡(2x)2+(x+sin⁡(2x)2)(ex+1)(ex+1)2cos⁡2(x)\frac{- \frac{x e^{x} \sin{\left(2 x \right)}}{2} + \left(x + \frac{\sin{\left(2 x \right)}}{2}\right) \left(e^{x} + 1\right)}{\left(e^{x} + 1\right)^{2} \cos^{2}{\left(x \right)}}

Gráfica
02468-8-6-4-2-1010-50005000
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Trazar el gráfico f'(x)
Primera derivada [src] 
  /       2   \               x       
x*\1 + tan (x)/ + tan(x)   x*e *tan(x)
------------------------ - -----------
              x                     2 
         1 + E              /     x\  
                            \1 + E /
−xextan⁡(x)(ex+1)2+x(tan⁡2(x)+1)+tan⁡(x)ex+1- \frac{x e^{x} \tan{\left(x \right)}}{\left(e^{x} + 1\right)^{2}} + \frac{x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}}{e^{x} + 1}
Simplificar
Segunda derivada [src] 
                                                                               /        x \          
                                                                               |     2*e  |  x       
                                                                             x*|1 - ------|*e *tan(x)
                  /  /       2   \         \  x                                |         x|          
         2      2*\x*\1 + tan (x)/ + tan(x)/*e        /       2   \            \    1 + e /          
2 + 2*tan (x) - ------------------------------- + 2*x*\1 + tan (x)/*tan(x) - ------------------------
                                  x                                                        x         
                             1 + e                                                    1 + e          
-----------------------------------------------------------------------------------------------------
                                                     x                                               
                                                1 + e
−x(1−2exex+1)extan⁡(x)ex+1+2x(tan⁡2(x)+1)tan⁡(x)−2(x(tan⁡2(x)+1)+tan⁡(x))exex+1+2tan⁡2(x)+2ex+1\frac{- \frac{x \left(1 - \frac{2 e^{x}}{e^{x} + 1}\right) e^{x} \tan{\left(x \right)}}{e^{x} + 1} + 2 x \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{2 \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right) e^{x}}{e^{x} + 1} + 2 \tan^{2}{\left(x \right)} + 2}{e^{x} + 1}
Simplificar
Tercera derivada [src] 
                                                                                                                                                /        x         2*x \          
                                                                                                 /        x \                                   |     6*e       6*e    |  x       
                                                                                                 |     2*e  | /  /       2   \         \  x   x*|1 - ------ + ---------|*e *tan(x)
                                                                                               3*|1 - ------|*\x*\1 + tan (x)/ + tan(x)/*e      |         x           2|          
                                                   /       2        /       2   \       \  x     |         x|                                   |    1 + e    /     x\ |          
  /       2   \ /             /         2   \\   6*\1 + tan (x) + x*\1 + tan (x)/*tan(x)/*e      \    1 + e /                                   \             \1 + e / /          
2*\1 + tan (x)/*\3*tan(x) + x*\1 + 3*tan (x)// - ------------------------------------------- - -------------------------------------------- - ------------------------------------
                                                                         x                                             x                                          x               
                                                                    1 + e                                         1 + e                                      1 + e                
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                           x                                                                                      
                                                                                      1 + e
−x(1−6exex+1+6e2x(ex+1)2)extan⁡(x)ex+1−3(1−2exex+1)(x(tan⁡2(x)+1)+tan⁡(x))exex+1+2(x(3tan⁡2(x)+1)+3tan⁡(x))(tan⁡2(x)+1)−6(x(tan⁡2(x)+1)tan⁡(x)+tan⁡2(x)+1)exex+1ex+1\frac{- \frac{x \left(1 - \frac{6 e^{x}}{e^{x} + 1} + \frac{6 e^{2 x}}{\left(e^{x} + 1\right)^{2}}\right) e^{x} \tan{\left(x \right)}}{e^{x} + 1} - \frac{3 \left(1 - \frac{2 e^{x}}{e^{x} + 1}\right) \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right) e^{x}}{e^{x} + 1} + 2 \left(x \left(3 \tan^{2}{\left(x \right)} + 1\right) + 3 \tan{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) - \frac{6 \left(x \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \tan^{2}{\left(x \right)} + 1\right) e^{x}}{e^{x} + 1}}{e^{x} + 1}
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Gráfico
Derivada de (x*tanx)/(1+e^x)
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