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Derivada de е^(x^2*tgx)

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

Ha introducido [src]
  2       
 x *tan(x)
E         
ex2tan(x)e^{x^{2} \tan{\left(x \right)}}
E^(x^2*tan(x))
Solución detallada
  1. Sustituimos u=x2tan(x)u = x^{2} \tan{\left(x \right)}.

  2. Derivado eue^{u} es.

  3. Luego se aplica una cadena de reglas. Multiplicamos por ddxx2tan(x)\frac{d}{d x} x^{2} \tan{\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)=x2f{\left(x \right)} = x^{2}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. Según el principio, aplicamos: x2x^{2} tenemos 2x2 x

      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:

        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: x2(sin2(x)+cos2(x))cos2(x)+2xtan(x)\frac{x^{2} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + 2 x \tan{\left(x \right)}

    Como resultado de la secuencia de reglas:

    (x2(sin2(x)+cos2(x))cos2(x)+2xtan(x))ex2tan(x)\left(\frac{x^{2} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + 2 x \tan{\left(x \right)}\right) e^{x^{2} \tan{\left(x \right)}}

  4. Simplificamos:

    x(x+sin(2x))ex2tan(x)cos2(x)\frac{x \left(x + \sin{\left(2 x \right)}\right) e^{x^{2} \tan{\left(x \right)}}}{\cos^{2}{\left(x \right)}}


Respuesta:

x(x+sin(2x))ex2tan(x)cos2(x)\frac{x \left(x + \sin{\left(2 x \right)}\right) e^{x^{2} \tan{\left(x \right)}}}{\cos^{2}{\left(x \right)}}

Primera derivada [src]
                                  2       
/ 2 /       2   \             \  x *tan(x)
\x *\1 + tan (x)/ + 2*x*tan(x)/*e         
(x2(tan2(x)+1)+2xtan(x))ex2tan(x)\left(x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) + 2 x \tan{\left(x \right)}\right) e^{x^{2} \tan{\left(x \right)}}
Segunda derivada [src]
/                                          2                                                \   2       
|            2 /             /       2   \\        /       2   \      2 /       2   \       |  x *tan(x)
\2*tan(x) + x *\2*tan(x) + x*\1 + tan (x)//  + 4*x*\1 + tan (x)/ + 2*x *\1 + tan (x)/*tan(x)/*e         
(x2(x(tan2(x)+1)+2tan(x))2+2x2(tan2(x)+1)tan(x)+4x(tan2(x)+1)+2tan(x))ex2tan(x)\left(x^{2} \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + 2 \tan{\left(x \right)}\right)^{2} + 2 x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 4 x \left(\tan^{2}{\left(x \right)} + 1\right) + 2 \tan{\left(x \right)}\right) e^{x^{2} \tan{\left(x \right)}}
Tercera derivada [src]
/                                               3                     2                                                                                                                                                   \   2       
|         2       3 /             /       2   \\       2 /       2   \       2    2    /       2   \       /             /       2   \\ /    /       2   \    2 /       2   \                \        /       2   \       |  x *tan(x)
\6 + 6*tan (x) + x *\2*tan(x) + x*\1 + tan (x)//  + 2*x *\1 + tan (x)/  + 4*x *tan (x)*\1 + tan (x)/ + 6*x*\2*tan(x) + x*\1 + tan (x)//*\2*x*\1 + tan (x)/ + x *\1 + tan (x)/*tan(x) + tan(x)/ + 12*x*\1 + tan (x)/*tan(x)/*e         
(x3(x(tan2(x)+1)+2tan(x))3+2x2(tan2(x)+1)2+4x2(tan2(x)+1)tan2(x)+6x(x(tan2(x)+1)+2tan(x))(x2(tan2(x)+1)tan(x)+2x(tan2(x)+1)+tan(x))+12x(tan2(x)+1)tan(x)+6tan2(x)+6)ex2tan(x)\left(x^{3} \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + 2 \tan{\left(x \right)}\right)^{3} + 2 x^{2} \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 4 x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 6 x \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + 2 \tan{\left(x \right)}\right) \left(x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 2 x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right) + 12 x \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 6 \tan^{2}{\left(x \right)} + 6\right) e^{x^{2} \tan{\left(x \right)}}