Sr Examen

Otras calculadoras


y=(e^x+1)/(e^x-tgx)

Derivada de y=(e^x+1)/(e^x-tgx)

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

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src]
    x      
   E  + 1  
-----------
 x         
E  - tan(x)
ex+1extan(x)\frac{e^{x} + 1}{e^{x} - \tan{\left(x \right)}}
(E^x + 1)/(E^x - tan(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)=ex+1f{\left(x \right)} = e^{x} + 1 y g(x)=extan(x)g{\left(x \right)} = e^{x} - \tan{\left(x \right)}.

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

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

    1. diferenciamos extan(x)e^{x} - \tan{\left(x \right)} miembro por miembro:

      1. La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.

        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:

          sin2(x)+cos2(x)cos2(x)\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

        Entonces, como resultado: sin2(x)+cos2(x)cos2(x)- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

      2. Derivado exe^{x} es.

      Como resultado de: sin2(x)+cos2(x)cos2(x)+ex- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + e^{x}

    Ahora aplicamos la regla de la derivada de una divesión:

    (sin2(x)+cos2(x)cos2(x)+ex)(ex+1)+(extan(x))ex(extan(x))2\frac{- \left(- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + e^{x}\right) \left(e^{x} + 1\right) + \left(e^{x} - \tan{\left(x \right)}\right) e^{x}}{\left(e^{x} - \tan{\left(x \right)}\right)^{2}}

  2. Simplificamos:

    2exsin(2x+π4)+ex+2(extan(x))2(cos(2x)+1)\frac{- \sqrt{2} e^{x} \sin{\left(2 x + \frac{\pi}{4} \right)} + e^{x} + 2}{\left(e^{x} - \tan{\left(x \right)}\right)^{2} \left(\cos{\left(2 x \right)} + 1\right)}


Respuesta:

2exsin(2x+π4)+ex+2(extan(x))2(cos(2x)+1)\frac{- \sqrt{2} e^{x} \sin{\left(2 x + \frac{\pi}{4} \right)} + e^{x} + 2}{\left(e^{x} - \tan{\left(x \right)}\right)^{2} \left(\cos{\left(2 x \right)} + 1\right)}

Gráfica
02468-8-6-4-2-1010-1000010000
Primera derivada [src]
      x       / x    \ /       2       x\
     e        \E  + 1/*\1 + tan (x) - E /
----------- + ---------------------------
 x                                2      
E  - tan(x)          / x         \       
                     \E  - tan(x)/       
(ex+1)(ex+tan2(x)+1)(extan(x))2+exextan(x)\frac{\left(e^{x} + 1\right) \left(- e^{x} + \tan^{2}{\left(x \right)} + 1\right)}{\left(e^{x} - \tan{\left(x \right)}\right)^{2}} + \frac{e^{x}}{e^{x} - \tan{\left(x \right)}}
Segunda derivada [src]
         /                           2                         \                               
         |         /       2       x\                          |                               
/     x\ |   x   2*\1 + tan (x) - e /      /       2   \       |                               
\1 + e /*|- e  + --------------------- + 2*\1 + tan (x)/*tan(x)|                               
         |                       x                             |     /       2       x\  x     
         \            -tan(x) + e                              /   2*\1 + tan (x) - e /*e     x
---------------------------------------------------------------- + ----------------------- + e 
                                     x                                              x          
                          -tan(x) + e                                    -tan(x) + e           
-----------------------------------------------------------------------------------------------
                                                     x                                         
                                          -tan(x) + e                                          
(ex+1)(2(tan2(x)+1)tan(x)ex+2(ex+tan2(x)+1)2extan(x))extan(x)+ex+2(ex+tan2(x)+1)exextan(x)extan(x)\frac{\frac{\left(e^{x} + 1\right) \left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - e^{x} + \frac{2 \left(- e^{x} + \tan^{2}{\left(x \right)} + 1\right)^{2}}{e^{x} - \tan{\left(x \right)}}\right)}{e^{x} - \tan{\left(x \right)}} + e^{x} + \frac{2 \left(- e^{x} + \tan^{2}{\left(x \right)} + 1\right) e^{x}}{e^{x} - \tan{\left(x \right)}}}{e^{x} - \tan{\left(x \right)}}
Tercera derivada [src]
         /                                                                        3                                                       \                                                                                              
         |                      2                               /       2       x\      /   x     /       2   \       \ /       2       x\|                               /                           2                         \        
/     x\ |   x     /       2   \         2    /       2   \   6*\1 + tan (x) - e /    6*\- e  + 2*\1 + tan (x)/*tan(x)/*\1 + tan (x) - e /|                               |         /       2       x\                          |        
\1 + e /*|- e  + 2*\1 + tan (x)/  + 4*tan (x)*\1 + tan (x)/ + --------------------- + ----------------------------------------------------|                               |   x   2*\1 + tan (x) - e /      /       2   \       |  x     
         |                                                                     2                                     x                    |                             3*|- e  + --------------------- + 2*\1 + tan (x)/*tan(x)|*e      
         |                                                       /           x\                           -tan(x) + e                     |     /       2       x\  x     |                       x                             |        
         \                                                       \-tan(x) + e /                                                           /   3*\1 + tan (x) - e /*e      \            -tan(x) + e                              /       x
------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------- + ------------------------------------------------------------ + e 
                                                                           x                                                                                   x                                           x                             
                                                                -tan(x) + e                                                                         -tan(x) + e                                 -tan(x) + e                              
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                          x                                                                                                              
                                                                                                               -tan(x) + e                                                                                                               
(ex+1)(6(2(tan2(x)+1)tan(x)ex)(ex+tan2(x)+1)extan(x)+2(tan2(x)+1)2+4(tan2(x)+1)tan2(x)ex+6(ex+tan2(x)+1)3(extan(x))2)extan(x)+ex+3(2(tan2(x)+1)tan(x)ex+2(ex+tan2(x)+1)2extan(x))exextan(x)+3(ex+tan2(x)+1)exextan(x)extan(x)\frac{\frac{\left(e^{x} + 1\right) \left(\frac{6 \left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - e^{x}\right) \left(- e^{x} + \tan^{2}{\left(x \right)} + 1\right)}{e^{x} - \tan{\left(x \right)}} + 2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} - e^{x} + \frac{6 \left(- e^{x} + \tan^{2}{\left(x \right)} + 1\right)^{3}}{\left(e^{x} - \tan{\left(x \right)}\right)^{2}}\right)}{e^{x} - \tan{\left(x \right)}} + e^{x} + \frac{3 \left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - e^{x} + \frac{2 \left(- e^{x} + \tan^{2}{\left(x \right)} + 1\right)^{2}}{e^{x} - \tan{\left(x \right)}}\right) e^{x}}{e^{x} - \tan{\left(x \right)}} + \frac{3 \left(- e^{x} + \tan^{2}{\left(x \right)} + 1\right) e^{x}}{e^{x} - \tan{\left(x \right)}}}{e^{x} - \tan{\left(x \right)}}
Gráfico
Derivada de y=(e^x+1)/(e^x-tgx)