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Derivada de la función/ x*exp/ x^2tgx/ x*exp(-x^2tgxx)

Derivada de x*exp(-x^2tgxx)

Función f() - derivada -er orden en el punto
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Ha introducido [src] 
     2         
   -x *tan(x)*x
x*e
xex−x2tan⁡(x)x e^{x - x^{2} \tan{\left(x \right)}} 
x*exp(((-x^2)*tan(x))*x)
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)=ex−x2tan⁡(x)g{\left(x \right)} = e^{x - x^{2} \tan{\left(x \right)}}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Sustituimos u=x−x2tan⁡(x)u = x - x^{2} \tan{\left(x \right)}.

    2. Derivado eue^{u} es.

    3. Luego se aplica una cadena de reglas. Multiplicamos por ddxx−x2tan⁡(x)\frac{d}{d x} 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)=−x2tan⁡(x)f{\left(x \right)} = - x^{2} \tan{\left(x \right)}; calculamos 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)=−x2f{\left(x \right)} = - x^{2}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

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

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

            Entonces, como resultado: −2x- 2 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:

            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: −x2(sin⁡2(x)+cos⁡2(x))cos⁡2(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)}

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

        1. Según el principio, aplicamos: xx tenemos 11

        Como resultado de: x(−x2(sin⁡2(x)+cos⁡2(x))cos⁡2(x)−2xtan⁡(x))+−x2tan⁡(x)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) + - x^{2} \tan{\left(x \right)}

      Como resultado de la secuencia de reglas:

      (x(−x2(sin⁡2(x)+cos⁡2(x))cos⁡2(x)−2xtan⁡(x))+−x2tan⁡(x))ex−x2tan⁡(x)\left(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) + - x^{2} \tan{\left(x \right)}\right) e^{x - x^{2} \tan{\left(x \right)}}

    Como resultado de: x(x(−x2(sin⁡2(x)+cos⁡2(x))cos⁡2(x)−2xtan⁡(x))+−x2tan⁡(x))ex−x2tan⁡(x)+ex−x2tan⁡(x)x \left(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) + - x^{2} \tan{\left(x \right)}\right) e^{x - x^{2} \tan{\left(x \right)}} + e^{x - x^{2} \tan{\left(x \right)}}

  2. Simplificamos:

    (−x3(x+3sin⁡(2x)2)+cos⁡2(x))e−x3tan⁡(x)cos⁡2(x)\frac{\left(- x^{3} \left(x + \frac{3 \sin{\left(2 x \right)}}{2}\right) + \cos^{2}{\left(x \right)}\right) e^{- x^{3} \tan{\left(x \right)}}}{\cos^{2}{\left(x \right)}}

Respuesta:

(−x3(x+3sin⁡(2x)2)+cos⁡2(x))e−x3tan⁡(x)cos⁡2(x)\frac{\left(- x^{3} \left(x + \frac{3 \sin{\left(2 x \right)}}{2}\right) + \cos^{2}{\left(x \right)}\right) e^{- x^{3} \tan{\left(x \right)}}}{\cos^{2}{\left(x \right)}}

Primera derivada [src] 
                                                        2               2         
  /  /   2 /       2   \             \     2       \  -x *tan(x)*x    -x *tan(x)*x
x*\x*\- x *\1 + tan (x)/ - 2*x*tan(x)/ + -x *tan(x)/*e             + e
x(x(−x2(tan⁡2(x)+1)−2xtan⁡(x))+−x2tan⁡(x))ex−x2tan⁡(x)+ex−x2tan⁡(x)x \left(x \left(- x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) - 2 x \tan{\left(x \right)}\right) + - x^{2} \tan{\left(x \right)}\right) e^{x - x^{2} \tan{\left(x \right)}} + e^{x - x^{2} \tan{\left(x \right)}}
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Segunda derivada [src] 
    /                                           2                                                \    3       
  2 |             3 /             /       2   \\        /       2   \      2 /       2   \       |  -x *tan(x)
-x *\12*tan(x) - x *\3*tan(x) + x*\1 + tan (x)//  + 8*x*\1 + tan (x)/ + 2*x *\1 + tan (x)/*tan(x)/*e
−x2(−x3(x(tan⁡2(x)+1)+3tan⁡(x))2+2x2(tan⁡2(x)+1)tan⁡(x)+8x(tan⁡2(x)+1)+12tan⁡(x))e−x3tan⁡(x)- x^{2} \left(- x^{3} \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + 3 \tan{\left(x \right)}\right)^{2} + 2 x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 8 x \left(\tan^{2}{\left(x \right)} + 1\right) + 12 \tan{\left(x \right)}\right) e^{- x^{3} \tan{\left(x \right)}}
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Tercera derivada [src] 
   /                                           3                                    2       /                                2                                                        \                                                                                                                                               \    3       
   |             6 /             /       2   \\       3 /             /       2   \\        |         2       2 /       2   \       2    2    /       2   \       /       2   \       |        /       2   \      3 /             /       2   \\ /               /       2   \    2 /       2   \       \       2 /       2   \       |  -x *tan(x)
-x*\24*tan(x) + x *\3*tan(x) + x*\1 + tan (x)//  - 3*x *\3*tan(x) + x*\1 + tan (x)//  + 2*x*\3 + 3*tan (x) + x *\1 + tan (x)/  + 2*x *tan (x)*\1 + tan (x)/ + 6*x*\1 + tan (x)/*tan(x)/ + 30*x*\1 + tan (x)/ - 6*x *\3*tan(x) + x*\1 + tan (x)//*\3*tan(x) + 3*x*\1 + tan (x)/ + x *\1 + tan (x)/*tan(x)/ + 12*x *\1 + tan (x)/*tan(x)/*e
−x(x6(x(tan⁡2(x)+1)+3tan⁡(x))3−3x3(x(tan⁡2(x)+1)+3tan⁡(x))2−6x3(x(tan⁡2(x)+1)+3tan⁡(x))(x2(tan⁡2(x)+1)tan⁡(x)+3x(tan⁡2(x)+1)+3tan⁡(x))+12x2(tan⁡2(x)+1)tan⁡(x)+30x(tan⁡2(x)+1)+2x(x2(tan⁡2(x)+1)2+2x2(tan⁡2(x)+1)tan⁡2(x)+6x(tan⁡2(x)+1)tan⁡(x)+3tan⁡2(x)+3)+24tan⁡(x))e−x3tan⁡(x)- x \left(x^{6} \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + 3 \tan{\left(x \right)}\right)^{3} - 3 x^{3} \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + 3 \tan{\left(x \right)}\right)^{2} - 6 x^{3} \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + 3 \tan{\left(x \right)}\right) \left(x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 3 x \left(\tan^{2}{\left(x \right)} + 1\right) + 3 \tan{\left(x \right)}\right) + 12 x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 30 x \left(\tan^{2}{\left(x \right)} + 1\right) + 2 x \left(x^{2} \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 2 x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 6 x \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 3 \tan^{2}{\left(x \right)} + 3\right) + 24 \tan{\left(x \right)}\right) e^{- x^{3} \tan{\left(x \right)}}
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