Sr Examen

Otras calculadoras

Derivada de la función/ tgx^2/ e^tgx/ x*e^tgx^2

Derivada de x*e^tgx^2

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

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
      2   
   tan (x)
x*E
etan⁡2(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)=etan⁡2(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=tan⁡2(x)u = \tan^{2}{\left(x \right)}.

    2. Derivado eue^{u} es.

    3. Luego se aplica una cadena de reglas. Multiplicamos por ddxtan⁡2(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:

          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 la secuencia de reglas:

        2(sin⁡2(x)+cos⁡2(x))tan⁡(x)cos⁡2(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(sin⁡2(x)+cos⁡2(x))etan⁡2(x)tan⁡(x)cos⁡2(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: etan⁡2(x)+2x(sin⁡2(x)+cos⁡2(x))etan⁡2(x)tan⁡(x)cos⁡2(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)cos⁡3(x)+1)etan⁡2(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)cos⁡3(x)+1)etan⁡2(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)
etan⁡2(x)+x(2tan⁡2(x)+2)etan⁡2(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)}
Simplificar
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(tan⁡2(x)+1)tan⁡2(x)+3tan⁡2(x)+1)+2tan⁡(x))(tan⁡2(x)+1)etan⁡2(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)}}
Simplificar
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(tan⁡2(x)+1)(2x(2(tan⁡2(x)+1)2tan⁡2(x)+3(tan⁡2(x)+1)2+6(tan⁡2(x)+1)tan⁡2(x)+6tan⁡2(x)+4)tan⁡(x)+6(tan⁡2(x)+1)tan⁡2(x)+9tan⁡2(x)+3)etan⁡2(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)}}
Simplificar
    © Sr Examen – Калькулятор