Sr Examen

Derivada de x*sinx*tgx

Función f() - derivada -er orden en el punto
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Gráfico:

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Solución

Ha introducido [src]
x*sin(x)*tan(x)
xsin(x)tan(x)x \sin{\left(x \right)} \tan{\left(x \right)}
(x*sin(x))*tan(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)=xsin(x)f{\left(x \right)} = x \sin{\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)=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)=sin(x)g{\left(x \right)} = \sin{\left(x \right)}; calculamos ddxg(x)\frac{d}{d x} g{\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)}

      Como resultado de: xcos(x)+sin(x)x \cos{\left(x \right)} + \sin{\left(x \right)}

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

  2. Simplificamos:

    (x+xcos2(x)+tan(x))sin(x)\left(x + \frac{x}{\cos^{2}{\left(x \right)}} + \tan{\left(x \right)}\right) \sin{\left(x \right)}


Respuesta:

(x+xcos2(x)+tan(x))sin(x)\left(x + \frac{x}{\cos^{2}{\left(x \right)}} + \tan{\left(x \right)}\right) \sin{\left(x \right)}

Gráfica
02468-8-6-4-2-1010-50005000
Primera derivada [src]
                               /       2   \       
(x*cos(x) + sin(x))*tan(x) + x*\1 + tan (x)/*sin(x)
x(tan2(x)+1)sin(x)+(xcos(x)+sin(x))tan(x)x \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} + \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \tan{\left(x \right)}
Segunda derivada [src]
                                   /       2   \                           /       2   \              
-(-2*cos(x) + x*sin(x))*tan(x) + 2*\1 + tan (x)/*(x*cos(x) + sin(x)) + 2*x*\1 + tan (x)/*sin(x)*tan(x)
2x(tan2(x)+1)sin(x)tan(x)(xsin(x)2cos(x))tan(x)+2(xcos(x)+sin(x))(tan2(x)+1)2 x \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \tan{\left(x \right)} - \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \tan{\left(x \right)} + 2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right)
Tercera derivada [src]
                                  /       2   \                            /       2   \                                  /       2   \ /         2   \       
-(3*sin(x) + x*cos(x))*tan(x) - 3*\1 + tan (x)/*(-2*cos(x) + x*sin(x)) + 6*\1 + tan (x)/*(x*cos(x) + sin(x))*tan(x) + 2*x*\1 + tan (x)/*\1 + 3*tan (x)/*sin(x)
2x(tan2(x)+1)(3tan2(x)+1)sin(x)3(xsin(x)2cos(x))(tan2(x)+1)+6(xcos(x)+sin(x))(tan2(x)+1)tan(x)(xcos(x)+3sin(x))tan(x)2 x \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} - 3 \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) + 6 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \left(x \cos{\left(x \right)} + 3 \sin{\left(x \right)}\right) \tan{\left(x \right)}
Gráfico
Derivada de x*sinx*tgx