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Derivada de x*e^tgx^2

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

Ha introducido [src]
      2   
   tan (x)
x*E       
etan2(x)xe^{\tan^{2}{\left(x \right)}} x
x*E^(tan(x)^2)
Solución detallada
  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)=etan2(x)g{\left(x \right)} = e^{\tan^{2}{\left(x \right)}}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Sustituimos u=tan2(x)u = \tan^{2}{\left(x \right)}.

    2. Derivado eue^{u} es.

    3. Luego se aplica una cadena de reglas. Multiplicamos por ddxtan2(x)\frac{d}{d x} \tan^{2}{\left(x \right)}:

      1. Sustituimos u=tan(x)u = \tan{\left(x \right)}.

      2. Según el principio, aplicamos: u2u^{2} tenemos 2u2 u

      3. Luego se aplica una cadena de reglas. Multiplicamos por ddxtan(x)\frac{d}{d x} \tan{\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:

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

        Como resultado de la secuencia de reglas:

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

      Como resultado de la secuencia de reglas:

      2(sin2(x)+cos2(x))etan2(x)tan(x)cos2(x)\frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) e^{\tan^{2}{\left(x \right)}} \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}}

    Como resultado de: etan2(x)+2x(sin2(x)+cos2(x))etan2(x)tan(x)cos2(x)e^{\tan^{2}{\left(x \right)}} + \frac{2 x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) e^{\tan^{2}{\left(x \right)}} \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}}

  2. Simplificamos:

    (2xsin(x)cos3(x)+1)etan2(x)\left(\frac{2 x \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}} + 1\right) e^{\tan^{2}{\left(x \right)}}


Respuesta:

(2xsin(x)cos3(x)+1)etan2(x)\left(\frac{2 x \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}} + 1\right) e^{\tan^{2}{\left(x \right)}}

Primera derivada [src]
    2                            2          
 tan (x)     /         2   \  tan (x)       
E        + x*\2 + 2*tan (x)/*e       *tan(x)
etan2(x)+x(2tan2(x)+2)etan2(x)tan(x)e^{\tan^{2}{\left(x \right)}} + x \left(2 \tan^{2}{\left(x \right)} + 2\right) e^{\tan^{2}{\left(x \right)}} \tan{\left(x \right)}
Segunda derivada [src]
                                                                             2   
  /       2   \ /             /         2           2    /       2   \\\  tan (x)
2*\1 + tan (x)/*\2*tan(x) + x*\1 + 3*tan (x) + 2*tan (x)*\1 + tan (x)///*e       
2(x(2(tan2(x)+1)tan2(x)+3tan2(x)+1)+2tan(x))(tan2(x)+1)etan2(x)2 \left(x \left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 3 \tan^{2}{\left(x \right)} + 1\right) + 2 \tan{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) e^{\tan^{2}{\left(x \right)}}
Tercera derivada [src]
                /                                              /                   2                              2                                  \       \     2   
  /       2   \ |         2           2    /       2   \       |      /       2   \         2        /       2   \     2           2    /       2   \|       |  tan (x)
2*\1 + tan (x)/*\3 + 9*tan (x) + 6*tan (x)*\1 + tan (x)/ + 2*x*\4 + 3*\1 + tan (x)/  + 6*tan (x) + 2*\1 + tan (x)/ *tan (x) + 6*tan (x)*\1 + tan (x)//*tan(x)/*e       
2(tan2(x)+1)(2x(2(tan2(x)+1)2tan2(x)+3(tan2(x)+1)2+6(tan2(x)+1)tan2(x)+6tan2(x)+4)tan(x)+6(tan2(x)+1)tan2(x)+9tan2(x)+3)etan2(x)2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(2 x \left(2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \tan^{2}{\left(x \right)} + 3 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 6 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 6 \tan^{2}{\left(x \right)} + 4\right) \tan{\left(x \right)} + 6 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 9 \tan^{2}{\left(x \right)} + 3\right) e^{\tan^{2}{\left(x \right)}}