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(xsinx)/(1+tgx)+xln^2x

Derivada de (xsinx)/(1+tgx)+xln^2x

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

Gráfico:

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Ha introducido [src]
 x*sin(x)         2   
---------- + x*log (x)
1 + tan(x)            
xlog(x)2+xsin(x)tan(x)+1x \log{\left(x \right)}^{2} + \frac{x \sin{\left(x \right)}}{\tan{\left(x \right)} + 1}
(x*sin(x))/(1 + tan(x)) + x*log(x)^2
Solución detallada
  1. diferenciamos xlog(x)2+xsin(x)tan(x)+1x \log{\left(x \right)}^{2} + \frac{x \sin{\left(x \right)}}{\tan{\left(x \right)} + 1} miembro por miembro:

    1. 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)=xsin(x)f{\left(x \right)} = x \sin{\left(x \right)} y g(x)=tan(x)+1g{\left(x \right)} = \tan{\left(x \right)} + 1.

      Para calcular 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)}

      Para calcular ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. diferenciamos tan(x)+1\tan{\left(x \right)} + 1 miembro por miembro:

        1. La derivada de una constante 11 es igual a cero.

        2. Reescribimos las funciones para diferenciar:

          tan(x)=sin(x)cos(x)\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}

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

      Ahora aplicamos la regla de la derivada de una divesión:

      x(sin2(x)+cos2(x))sin(x)cos2(x)+(xcos(x)+sin(x))(tan(x)+1)(tan(x)+1)2\frac{- \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) \left(\tan{\left(x \right)} + 1\right)}{\left(\tan{\left(x \right)} + 1\right)^{2}}

    2. 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)=log(x)2g{\left(x \right)} = \log{\left(x \right)}^{2}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

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

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

      3. Luego se aplica una cadena de reglas. Multiplicamos por ddxlog(x)\frac{d}{d x} \log{\left(x \right)}:

        1. Derivado log(x)\log{\left(x \right)} es 1x\frac{1}{x}.

        Como resultado de la secuencia de reglas:

        2log(x)x\frac{2 \log{\left(x \right)}}{x}

      Como resultado de: log(x)2+2log(x)\log{\left(x \right)}^{2} + 2 \log{\left(x \right)}

    Como resultado de: x(sin2(x)+cos2(x))sin(x)cos2(x)+(xcos(x)+sin(x))(tan(x)+1)(tan(x)+1)2+log(x)2+2log(x)\frac{- \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) \left(\tan{\left(x \right)} + 1\right)}{\left(\tan{\left(x \right)} + 1\right)^{2}} + \log{\left(x \right)}^{2} + 2 \log{\left(x \right)}

  2. Simplificamos:

    xsin(x)+(xcos(x)+sin(x))(tan(x)+1)cos2(x)+(log(x)+2)(tan(x)+1)2log(x)cos2(x)(tan(x)+1)2cos2(x)\frac{- x \sin{\left(x \right)} + \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \left(\tan{\left(x \right)} + 1\right) \cos^{2}{\left(x \right)} + \left(\log{\left(x \right)} + 2\right) \left(\tan{\left(x \right)} + 1\right)^{2} \log{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{2} \cos^{2}{\left(x \right)}}


Respuesta:

xsin(x)+(xcos(x)+sin(x))(tan(x)+1)cos2(x)+(log(x)+2)(tan(x)+1)2log(x)cos2(x)(tan(x)+1)2cos2(x)\frac{- x \sin{\left(x \right)} + \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \left(\tan{\left(x \right)} + 1\right) \cos^{2}{\left(x \right)} + \left(\log{\left(x \right)} + 2\right) \left(\tan{\left(x \right)} + 1\right)^{2} \log{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{2} \cos^{2}{\left(x \right)}}

Gráfica
02468-8-6-4-2-1010-20002000
Primera derivada [src]
                                           /        2   \       
   2                 x*cos(x) + sin(x)   x*\-1 - tan (x)/*sin(x)
log (x) + 2*log(x) + ----------------- + -----------------------
                         1 + tan(x)                       2     
                                              (1 + tan(x))      
x(tan2(x)1)sin(x)(tan(x)+1)2+xcos(x)+sin(x)tan(x)+1+log(x)2+2log(x)\frac{x \left(- \tan^{2}{\left(x \right)} - 1\right) \sin{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{2}} + \frac{x \cos{\left(x \right)} + \sin{\left(x \right)}}{\tan{\left(x \right)} + 1} + \log{\left(x \right)}^{2} + 2 \log{\left(x \right)}
Segunda derivada [src]
                                                                                                                                           2                                         
                                      /       2   \                       /       2   \            /       2   \              /       2   \               /       2   \              
2   -2*cos(x) + x*sin(x)   2*log(x)   \1 + tan (x)/*(x*cos(x) + sin(x))   \1 + tan (x)/*sin(x)   x*\1 + tan (x)/*cos(x)   2*x*\1 + tan (x)/ *sin(x)   2*x*\1 + tan (x)/*sin(x)*tan(x)
- - -------------------- + -------- - --------------------------------- - -------------------- - ---------------------- + ------------------------- - -------------------------------
x        1 + tan(x)           x                             2                            2                       2                          3                              2         
                                                (1 + tan(x))                 (1 + tan(x))            (1 + tan(x))               (1 + tan(x))                   (1 + tan(x))          
2x(tan2(x)+1)sin(x)tan(x)(tan(x)+1)2x(tan2(x)+1)cos(x)(tan(x)+1)2+2x(tan2(x)+1)2sin(x)(tan(x)+1)3xsin(x)2cos(x)tan(x)+1(xcos(x)+sin(x))(tan2(x)+1)(tan(x)+1)2(tan2(x)+1)sin(x)(tan(x)+1)2+2log(x)x+2x- \frac{2 x \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \tan{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{2}} - \frac{x \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{2}} + \frac{2 x \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{3}} - \frac{x \sin{\left(x \right)} - 2 \cos{\left(x \right)}}{\tan{\left(x \right)} + 1} - \frac{\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{\left(\tan{\left(x \right)} + 1\right)^{2}} - \frac{\left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{2}} + \frac{2 \log{\left(x \right)}}{x} + \frac{2}{x}
Tercera derivada [src]
                                                                           2                                                                               2                                                    3                                                           2                                                                        2                                                                                                 2              
                                     /       2   \            /       2   \                          /       2   \                            /       2   \             /       2   \              /       2   \             /       2   \                     /       2   \             /       2   \                                  /       2   \                  2    /       2   \              /       2   \                      /       2   \               
  3*sin(x) + x*cos(x)   2*log(x)   2*\1 + tan (x)/*cos(x)   2*\1 + tan (x)/ *(x*cos(x) + sin(x))   2*\1 + tan (x)/*(-2*cos(x) + x*sin(x))   4*\1 + tan (x)/ *sin(x)   x*\1 + tan (x)/*sin(x)   6*x*\1 + tan (x)/ *sin(x)   4*\1 + tan (x)/*sin(x)*tan(x)   2*x*\1 + tan (x)/ *sin(x)   2*\1 + tan (x)/*(x*cos(x) + sin(x))*tan(x)   4*x*\1 + tan (x)/ *cos(x)   4*x*tan (x)*\1 + tan (x)/*sin(x)   4*x*\1 + tan (x)/*cos(x)*tan(x)   12*x*\1 + tan (x)/ *sin(x)*tan(x)
- ------------------- - -------- - ---------------------- + ------------------------------------ + -------------------------------------- + ----------------------- + ---------------------- - ------------------------- - ----------------------------- - ------------------------- - ------------------------------------------ + ------------------------- - -------------------------------- - ------------------------------- + ---------------------------------
       1 + tan(x)           2                      2                               3                                       2                                 3                        2                          4                             2                             2                                   2                                    3                              2                                  2                                  3          
                           x           (1 + tan(x))                    (1 + tan(x))                            (1 + tan(x))                      (1 + tan(x))             (1 + tan(x))               (1 + tan(x))                  (1 + tan(x))                  (1 + tan(x))                        (1 + tan(x))                         (1 + tan(x))                   (1 + tan(x))                       (1 + tan(x))                       (1 + tan(x))           
2x(tan2(x)+1)2sin(x)(tan(x)+1)24x(tan2(x)+1)sin(x)tan2(x)(tan(x)+1)2+x(tan2(x)+1)sin(x)(tan(x)+1)24x(tan2(x)+1)cos(x)tan(x)(tan(x)+1)2+12x(tan2(x)+1)2sin(x)tan(x)(tan(x)+1)3+4x(tan2(x)+1)2cos(x)(tan(x)+1)36x(tan2(x)+1)3sin(x)(tan(x)+1)4+2(xsin(x)2cos(x))(tan2(x)+1)(tan(x)+1)22(xcos(x)+sin(x))(tan2(x)+1)tan(x)(tan(x)+1)2+2(xcos(x)+sin(x))(tan2(x)+1)2(tan(x)+1)3xcos(x)+3sin(x)tan(x)+14(tan2(x)+1)sin(x)tan(x)(tan(x)+1)22(tan2(x)+1)cos(x)(tan(x)+1)2+4(tan2(x)+1)2sin(x)(tan(x)+1)32log(x)x2- \frac{2 x \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{2}} - \frac{4 x \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \tan^{2}{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{2}} + \frac{x \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{2}} - \frac{4 x \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} \tan{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{2}} + \frac{12 x \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin{\left(x \right)} \tan{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{3}} + \frac{4 x \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \cos{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{3}} - \frac{6 x \left(\tan^{2}{\left(x \right)} + 1\right)^{3} \sin{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{4}} + \frac{2 \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{\left(\tan{\left(x \right)} + 1\right)^{2}} - \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{2}} + \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\left(\tan{\left(x \right)} + 1\right)^{3}} - \frac{x \cos{\left(x \right)} + 3 \sin{\left(x \right)}}{\tan{\left(x \right)} + 1} - \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \tan{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{2}} - \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{2}} + \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{3}} - \frac{2 \log{\left(x \right)}}{x^{2}}
Gráfico
Derivada de (xsinx)/(1+tgx)+xln^2x