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xln(sinx)/sqrt(x^2+4)+sqrt(x^2+4)cosx/sinx

Derivada de xln(sinx)/sqrt(x^2+4)+sqrt(x^2+4)cosx/sinx

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

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Solución

Ha introducido [src]
                   ________       
                  /  2            
x*log(sin(x))   \/  x  + 4 *cos(x)
------------- + ------------------
    ________          sin(x)      
   /  2                           
 \/  x  + 4                       
xlog(sin(x))x2+4+x2+4cos(x)sin(x)\frac{x \log{\left(\sin{\left(x \right)} \right)}}{\sqrt{x^{2} + 4}} + \frac{\sqrt{x^{2} + 4} \cos{\left(x \right)}}{\sin{\left(x \right)}}
(x*log(sin(x)))/sqrt(x^2 + 4) + (sqrt(x^2 + 4)*cos(x))/sin(x)
Solución detallada
  1. diferenciamos xlog(sin(x))x2+4+x2+4cos(x)sin(x)\frac{x \log{\left(\sin{\left(x \right)} \right)}}{\sqrt{x^{2} + 4}} + \frac{\sqrt{x^{2} + 4} \cos{\left(x \right)}}{\sin{\left(x \right)}} 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)=xlog(sin(x))f{\left(x \right)} = x \log{\left(\sin{\left(x \right)} \right)} y g(x)=x2+4g{\left(x \right)} = \sqrt{x^{2} + 4}.

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

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

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

        3. Luego se aplica una cadena de reglas. Multiplicamos por ddxsin(x)\frac{d}{d x} \sin{\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 la secuencia de reglas:

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

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

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

      1. Sustituimos u=x2+4u = x^{2} + 4.

      2. Según el principio, aplicamos: u\sqrt{u} tenemos 12u\frac{1}{2 \sqrt{u}}

      3. Luego se aplica una cadena de reglas. Multiplicamos por ddx(x2+4)\frac{d}{d x} \left(x^{2} + 4\right):

        1. diferenciamos x2+4x^{2} + 4 miembro por miembro:

          1. La derivada de una constante 44 es igual a cero.

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

          Como resultado de: 2x2 x

        Como resultado de la secuencia de reglas:

        xx2+4\frac{x}{\sqrt{x^{2} + 4}}

      Ahora aplicamos la regla de la derivada de una divesión:

      x2log(sin(x))x2+4+x2+4(xcos(x)sin(x)+log(sin(x)))x2+4\frac{- \frac{x^{2} \log{\left(\sin{\left(x \right)} \right)}}{\sqrt{x^{2} + 4}} + \sqrt{x^{2} + 4} \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right)}{x^{2} + 4}

    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)=x2+4cos(x)f{\left(x \right)} = \sqrt{x^{2} + 4} \cos{\left(x \right)} y g(x)=sin(x)g{\left(x \right)} = \sin{\left(x \right)}.

      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)=x2+4f{\left(x \right)} = \sqrt{x^{2} + 4}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

        1. Sustituimos u=x2+4u = x^{2} + 4.

        2. Según el principio, aplicamos: u\sqrt{u} tenemos 12u\frac{1}{2 \sqrt{u}}

        3. Luego se aplica una cadena de reglas. Multiplicamos por ddx(x2+4)\frac{d}{d x} \left(x^{2} + 4\right):

          1. diferenciamos x2+4x^{2} + 4 miembro por miembro:

            1. La derivada de una constante 44 es igual a cero.

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

            Como resultado de: 2x2 x

          Como resultado de la secuencia de reglas:

          xx2+4\frac{x}{\sqrt{x^{2} + 4}}

        g(x)=cos(x)g{\left(x \right)} = \cos{\left(x \right)}; calculamos 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)}

        Como resultado de: xcos(x)x2+4x2+4sin(x)\frac{x \cos{\left(x \right)}}{\sqrt{x^{2} + 4}} - \sqrt{x^{2} + 4} \sin{\left(x \right)}

      Para calcular 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)}

      Ahora aplicamos la regla de la derivada de una divesión:

      x2+4cos2(x)+(xcos(x)x2+4x2+4sin(x))sin(x)sin2(x)\frac{- \sqrt{x^{2} + 4} \cos^{2}{\left(x \right)} + \left(\frac{x \cos{\left(x \right)}}{\sqrt{x^{2} + 4}} - \sqrt{x^{2} + 4} \sin{\left(x \right)}\right) \sin{\left(x \right)}}{\sin^{2}{\left(x \right)}}

    Como resultado de: x2+4cos2(x)+(xcos(x)x2+4x2+4sin(x))sin(x)sin2(x)+x2log(sin(x))x2+4+x2+4(xcos(x)sin(x)+log(sin(x)))x2+4\frac{- \sqrt{x^{2} + 4} \cos^{2}{\left(x \right)} + \left(\frac{x \cos{\left(x \right)}}{\sqrt{x^{2} + 4}} - \sqrt{x^{2} + 4} \sin{\left(x \right)}\right) \sin{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{- \frac{x^{2} \log{\left(\sin{\left(x \right)} \right)}}{\sqrt{x^{2} + 4}} + \sqrt{x^{2} + 4} \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right)}{x^{2} + 4}

  2. Simplificamos:

    x4+x3sin(2x)8x2+4xsin(2x)2log(sin(x))cos(2x)+2log(sin(x))16(x2+4)32sin2(x)\frac{- x^{4} + x^{3} \sin{\left(2 x \right)} - 8 x^{2} + 4 x \sin{\left(2 x \right)} - 2 \log{\left(\sin{\left(x \right)} \right)} \cos{\left(2 x \right)} + 2 \log{\left(\sin{\left(x \right)} \right)} - 16}{\left(x^{2} + 4\right)^{\frac{3}{2}} \sin^{2}{\left(x \right)}}


Respuesta:

x4+x3sin(2x)8x2+4xsin(2x)2log(sin(x))cos(2x)+2log(sin(x))16(x2+4)32sin2(x)\frac{- x^{4} + x^{3} \sin{\left(2 x \right)} - 8 x^{2} + 4 x \sin{\left(2 x \right)} - 2 \log{\left(\sin{\left(x \right)} \right)} \cos{\left(2 x \right)} + 2 \log{\left(\sin{\left(x \right)} \right)} - 16}{\left(x^{2} + 4\right)^{\frac{3}{2}} \sin^{2}{\left(x \right)}}

Gráfica
02468-8-6-4-2-1010-5000050000
Primera derivada [src]
                              ________                                                            
                             /  2                 x*cos(x)                                        
                         - \/  x  + 4 *sin(x) + -----------                                       
x*cos(x)                                           ________                       ________        
-------- + log(sin(x))                            /  2         2                 /  2         2   
 sin(x)                                         \/  x  + 4    x *log(sin(x))   \/  x  + 4 *cos (x)
---------------------- + ---------------------------------- - -------------- - -------------------
        ________                       sin(x)                          3/2              2         
       /  2                                                    / 2    \              sin (x)      
     \/  x  + 4                                                \x  + 4/                           
x2log(sin(x))(x2+4)32x2+4cos2(x)sin2(x)+xcos(x)x2+4x2+4sin(x)sin(x)+xcos(x)sin(x)+log(sin(x))x2+4- \frac{x^{2} \log{\left(\sin{\left(x \right)} \right)}}{\left(x^{2} + 4\right)^{\frac{3}{2}}} - \frac{\sqrt{x^{2} + 4} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{\frac{x \cos{\left(x \right)}}{\sqrt{x^{2} + 4}} - \sqrt{x^{2} + 4} \sin{\left(x \right)}}{\sin{\left(x \right)}} + \frac{\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}}{\sqrt{x^{2} + 4}}
Segunda derivada [src]
                                ________                          2                                                      /     ________                     \                                                                                                                                      
                      2        /      2              cos(x)      x *cos(x)     2*x*sin(x)                                |    /      2             x*cos(x) |                                                                                                                                      
      2*cos(x)   x*cos (x)   \/  4 + x  *cos(x) - ----------- + ----------- + -----------                                |- \/  4 + x  *sin(x) + -----------|*cos(x)                                                                                                                               
  x - -------- + ---------                           ________           3/2      ________     /x*cos(x)              \   |                          ________|                                 ________                ________                                                                     
       sin(x)        2                              /      2    /     2\        /      2    x*|-------- + log(sin(x))|   |                         /      2 |                                /      2     3          /      2              3                         2                2            
                  sin (x)                         \/  4 + x     \4 + x /      \/  4 + x       \ sin(x)               /   \                       \/  4 + x  /          2*x*log(sin(x))   2*\/  4 + x  *cos (x)   2*\/  4 + x  *cos(x)   3*x *log(sin(x))        x*cos (x)            x *cos(x)     
- ------------------------ - ------------------------------------------------------------ - -------------------------- - ------------------------------------------- - --------------- + --------------------- + -------------------- + ---------------- - ------------------- - ------------------
           ________                                     sin(x)                                             3/2                                2                                  3/2               3                    sin(x)                    5/2         ________                   3/2       
          /      2                                                                                 /     2\                                sin (x)                       /     2\               sin (x)                                   /     2\           /      2     2      /     2\          
        \/  4 + x                                                                                  \4 + x /                                                              \4 + x /                                                         \4 + x /         \/  4 + x  *sin (x)   \4 + x /   *sin(x)
3x3log(sin(x))(x2+4)52x2cos(x)(x2+4)32sin(x)xcos2(x)x2+4sin2(x)x(xcos(x)sin(x)+log(sin(x)))(x2+4)322xlog(sin(x))(x2+4)32+2x2+4cos(x)sin(x)+2x2+4cos3(x)sin3(x)(xcos(x)x2+4x2+4sin(x))cos(x)sin2(x)x2cos(x)(x2+4)32+2xsin(x)x2+4+x2+4cos(x)cos(x)x2+4sin(x)x+xcos2(x)sin2(x)2cos(x)sin(x)x2+4\frac{3 x^{3} \log{\left(\sin{\left(x \right)} \right)}}{\left(x^{2} + 4\right)^{\frac{5}{2}}} - \frac{x^{2} \cos{\left(x \right)}}{\left(x^{2} + 4\right)^{\frac{3}{2}} \sin{\left(x \right)}} - \frac{x \cos^{2}{\left(x \right)}}{\sqrt{x^{2} + 4} \sin^{2}{\left(x \right)}} - \frac{x \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right)}{\left(x^{2} + 4\right)^{\frac{3}{2}}} - \frac{2 x \log{\left(\sin{\left(x \right)} \right)}}{\left(x^{2} + 4\right)^{\frac{3}{2}}} + \frac{2 \sqrt{x^{2} + 4} \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2 \sqrt{x^{2} + 4} \cos^{3}{\left(x \right)}}{\sin^{3}{\left(x \right)}} - \frac{\left(\frac{x \cos{\left(x \right)}}{\sqrt{x^{2} + 4}} - \sqrt{x^{2} + 4} \sin{\left(x \right)}\right) \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{\frac{x^{2} \cos{\left(x \right)}}{\left(x^{2} + 4\right)^{\frac{3}{2}}} + \frac{2 x \sin{\left(x \right)}}{\sqrt{x^{2} + 4}} + \sqrt{x^{2} + 4} \cos{\left(x \right)} - \frac{\cos{\left(x \right)}}{\sqrt{x^{2} + 4}}}{\sin{\left(x \right)}} - \frac{x + \frac{x \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}}{\sqrt{x^{2} + 4}}
Tercera derivada [src]
                                                                                 ________                           ________                                                       2             3                                                                                                                                                                                        /     ________                     \     /   ________                          2                      \                                                                                                                                                                      
                                          2             3                       /      2             x*cos(x)      /      2             3*sin(x)     3*x*cos(x)    3*x*cos(x)   3*x *sin(x)   3*x *cos(x)                                                                                                                                          /                    2   \        2    |    /      2             x*cos(x) |     |  /      2              cos(x)      x *cos(x)     2*x*sin(x)|                                                                                                                                                                      
                                     3*cos (x)   2*x*cos (x)   2*x*cos(x)   - \/  4 + x  *sin(x) + -----------   \/  4 + x  *sin(x) - ----------- - ----------- - ----------- + ----------- + -----------                                                                                                                                          |    2*cos(x)   x*cos (x)|   2*cos (x)*|- \/  4 + x  *sin(x) + -----------|   2*|\/  4 + x  *cos(x) - ----------- + ----------- + -----------|*cos(x)                                                                                                                                                               
                                -3 - --------- + ----------- + ----------                             ________                           ________           3/2      ________           3/2           5/2   x*cos(x)                                                                                ________                ________           2*x*|x - -------- + ---------|             |                          ________|     |                        ________           3/2      ________|             2 /x*cos(x)              \                                                                                                                               
       ________         2                2            3          sin(x)                              /      2                           /      2    /     2\        /      2    /     2\      /     2\      -------- + log(sin(x))                            2                4                   /      2     2          /      2     4          |     sin(x)        2    |             |                         /      2 |     |                       /      2    /     2\        /      2 |          3*x *|-------- + log(sin(x))|       2                                           2    2                    3                                     3           
      /      2         x              sin (x)      sin (x)                                         \/  4 + x                          \/  4 + x     \4 + x /      \/  4 + x     \4 + x /      \4 + x /       sin(x)                  2*log(sin(x))         cos (x)         15*x *log(sin(x))   8*\/  4 + x  *cos (x)   6*\/  4 + x  *cos (x)       \                sin (x) /             \                       \/  4 + x  /     \                     \/  4 + x     \4 + x /      \/  4 + x  /               \ sin(x)               /   15*x *log(sin(x))       4*x*cos(x)           2*x *cos (x)          4*x*cos (x)           4*x*cos(x)          6*x *cos(x)    
- 2*\/  4 + x   + ----------- + ----------------------------------------- + ---------------------------------- + ---------------------------------------------------------------------------------------- - ---------------------- - ------------- - ------------------- - ----------------- - --------------------- - --------------------- + ------------------------------ + ---------------------------------------------- + ----------------------------------------------------------------------- + ----------------------------- + ----------------- - ------------------ + ------------------- + ------------------- + ------------------ + ------------------
                          3/2                     ________                                sin(x)                                                          sin(x)                                                         3/2                  3/2       ________                      7/2                2                       4                              3/2                                   3                                                             2                                                       5/2                       5/2              3/2                  3/2              ________              ________                  5/2       
                  /     2\                       /      2                                                                                                                                                        /     2\             /     2\         /      2     2         /     2\                sin (x)                 sin (x)                   /     2\                                   sin (x)                                                       sin (x)                                            /     2\                  /     2\         /     2\             /     2\       2        /      2     3        /      2           /     2\          
                  \4 + x /                     \/  4 + x                                                                                                                                                         \4 + x /             \4 + x /       \/  4 + x  *sin (x)      \4 + x /                                                                  \4 + x /                                                                                                                                                    \4 + x /                  \4 + x /         \4 + x /   *sin(x)   \4 + x /   *sin (x)   \/  4 + x  *sin (x)   \/  4 + x  *sin(x)   \4 + x /   *sin(x)
15x4log(sin(x))(x2+4)72+6x3cos(x)(x2+4)52sin(x)+x2(x2+4)32+2x2cos2(x)(x2+4)32sin2(x)+3x2(xcos(x)sin(x)+log(sin(x)))(x2+4)52+15x2log(sin(x))(x2+4)52+4xcos(x)x2+4sin(x)+4xcos3(x)x2+4sin3(x)+2x(x+xcos2(x)sin2(x)2cos(x)sin(x))(x2+4)324xcos(x)(x2+4)32sin(x)2x2+48x2+4cos2(x)sin2(x)6x2+4cos4(x)sin4(x)+xcos(x)x2+4x2+4sin(x)sin(x)+2(xcos(x)x2+4x2+4sin(x))cos2(x)sin3(x)+2(x2cos(x)(x2+4)32+2xsin(x)x2+4+x2+4cos(x)cos(x)x2+4)cos(x)sin2(x)+3x3cos(x)(x2+4)52+3x2sin(x)(x2+4)323xcos(x)x2+43xcos(x)(x2+4)32+x2+4sin(x)3sin(x)x2+4sin(x)+2xcos(x)sin(x)+2xcos3(x)sin3(x)33cos2(x)sin2(x)x2+4cos2(x)x2+4sin2(x)xcos(x)sin(x)+log(sin(x))(x2+4)322log(sin(x))(x2+4)32- \frac{15 x^{4} \log{\left(\sin{\left(x \right)} \right)}}{\left(x^{2} + 4\right)^{\frac{7}{2}}} + \frac{6 x^{3} \cos{\left(x \right)}}{\left(x^{2} + 4\right)^{\frac{5}{2}} \sin{\left(x \right)}} + \frac{x^{2}}{\left(x^{2} + 4\right)^{\frac{3}{2}}} + \frac{2 x^{2} \cos^{2}{\left(x \right)}}{\left(x^{2} + 4\right)^{\frac{3}{2}} \sin^{2}{\left(x \right)}} + \frac{3 x^{2} \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right)}{\left(x^{2} + 4\right)^{\frac{5}{2}}} + \frac{15 x^{2} \log{\left(\sin{\left(x \right)} \right)}}{\left(x^{2} + 4\right)^{\frac{5}{2}}} + \frac{4 x \cos{\left(x \right)}}{\sqrt{x^{2} + 4} \sin{\left(x \right)}} + \frac{4 x \cos^{3}{\left(x \right)}}{\sqrt{x^{2} + 4} \sin^{3}{\left(x \right)}} + \frac{2 x \left(x + \frac{x \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right)}{\left(x^{2} + 4\right)^{\frac{3}{2}}} - \frac{4 x \cos{\left(x \right)}}{\left(x^{2} + 4\right)^{\frac{3}{2}} \sin{\left(x \right)}} - 2 \sqrt{x^{2} + 4} - \frac{8 \sqrt{x^{2} + 4} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{6 \sqrt{x^{2} + 4} \cos^{4}{\left(x \right)}}{\sin^{4}{\left(x \right)}} + \frac{\frac{x \cos{\left(x \right)}}{\sqrt{x^{2} + 4}} - \sqrt{x^{2} + 4} \sin{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2 \left(\frac{x \cos{\left(x \right)}}{\sqrt{x^{2} + 4}} - \sqrt{x^{2} + 4} \sin{\left(x \right)}\right) \cos^{2}{\left(x \right)}}{\sin^{3}{\left(x \right)}} + \frac{2 \left(\frac{x^{2} \cos{\left(x \right)}}{\left(x^{2} + 4\right)^{\frac{3}{2}}} + \frac{2 x \sin{\left(x \right)}}{\sqrt{x^{2} + 4}} + \sqrt{x^{2} + 4} \cos{\left(x \right)} - \frac{\cos{\left(x \right)}}{\sqrt{x^{2} + 4}}\right) \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{\frac{3 x^{3} \cos{\left(x \right)}}{\left(x^{2} + 4\right)^{\frac{5}{2}}} + \frac{3 x^{2} \sin{\left(x \right)}}{\left(x^{2} + 4\right)^{\frac{3}{2}}} - \frac{3 x \cos{\left(x \right)}}{\sqrt{x^{2} + 4}} - \frac{3 x \cos{\left(x \right)}}{\left(x^{2} + 4\right)^{\frac{3}{2}}} + \sqrt{x^{2} + 4} \sin{\left(x \right)} - \frac{3 \sin{\left(x \right)}}{\sqrt{x^{2} + 4}}}{\sin{\left(x \right)}} + \frac{\frac{2 x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2 x \cos^{3}{\left(x \right)}}{\sin^{3}{\left(x \right)}} - 3 - \frac{3 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}}{\sqrt{x^{2} + 4}} - \frac{\cos^{2}{\left(x \right)}}{\sqrt{x^{2} + 4} \sin^{2}{\left(x \right)}} - \frac{\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}}{\left(x^{2} + 4\right)^{\frac{3}{2}}} - \frac{2 \log{\left(\sin{\left(x \right)} \right)}}{\left(x^{2} + 4\right)^{\frac{3}{2}}}
Gráfico
Derivada de xln(sinx)/sqrt(x^2+4)+sqrt(x^2+4)cosx/sinx