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xln(sinx+sqrt(1+(sinx)^2))
Derivada de la función/ (sinx)/ xln(sinx+sqrt(1+(sinx)^2))

Derivada de xln(sinx+sqrt(1+(sinx)^2))

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

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Solución

Ha introducido [src] 
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x*log\sin(x) + \/  1 + sin (x) /
xlog⁡(sin⁡2(x)+1+sin⁡(x))x \log{\left(\sqrt{\sin^{2}{\left(x \right)} + 1} + \sin{\left(x \right)} \right)} 
x*log(sin(x) + sqrt(1 + sin(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)=log⁡(sin⁡2(x)+1+sin⁡(x))g{\left(x \right)} = \log{\left(\sqrt{\sin^{2}{\left(x \right)} + 1} + \sin{\left(x \right)} \right)}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Sustituimos u=sin⁡2(x)+1+sin⁡(x)u = \sqrt{\sin^{2}{\left(x \right)} + 1} + \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 ddx(sin⁡2(x)+1+sin⁡(x))\frac{d}{d x} \left(\sqrt{\sin^{2}{\left(x \right)} + 1} + \sin{\left(x \right)}\right):

      1. diferenciamos sin⁡2(x)+1+sin⁡(x)\sqrt{\sin^{2}{\left(x \right)} + 1} + \sin{\left(x \right)} miembro por miembro:

        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)}

        2. Sustituimos u=sin⁡2(x)+1u = \sin^{2}{\left(x \right)} + 1.

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

        4. Luego se aplica una cadena de reglas. Multiplicamos por ddx(sin⁡2(x)+1)\frac{d}{d x} \left(\sin^{2}{\left(x \right)} + 1\right):

          1. diferenciamos sin⁡2(x)+1\sin^{2}{\left(x \right)} + 1 miembro por miembro:

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

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

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

            4. 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:

              2sin⁡(x)cos⁡(x)2 \sin{\left(x \right)} \cos{\left(x \right)}

            Como resultado de: 2sin⁡(x)cos⁡(x)2 \sin{\left(x \right)} \cos{\left(x \right)}

          Como resultado de la secuencia de reglas:

          sin⁡(x)cos⁡(x)sin⁡2(x)+1\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\sqrt{\sin^{2}{\left(x \right)} + 1}}

        Como resultado de: cos⁡(x)+sin⁡(x)cos⁡(x)sin⁡2(x)+1\cos{\left(x \right)} + \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\sqrt{\sin^{2}{\left(x \right)} + 1}}

      Como resultado de la secuencia de reglas:

      cos⁡(x)+sin⁡(x)cos⁡(x)sin⁡2(x)+1sin⁡2(x)+1+sin⁡(x)\frac{\cos{\left(x \right)} + \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\sqrt{\sin^{2}{\left(x \right)} + 1}}}{\sqrt{\sin^{2}{\left(x \right)} + 1} + \sin{\left(x \right)}}

    Como resultado de: x(cos⁡(x)+sin⁡(x)cos⁡(x)sin⁡2(x)+1)sin⁡2(x)+1+sin⁡(x)+log⁡(sin⁡2(x)+1+sin⁡(x))\frac{x \left(\cos{\left(x \right)} + \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\sqrt{\sin^{2}{\left(x \right)} + 1}}\right)}{\sqrt{\sin^{2}{\left(x \right)} + 1} + \sin{\left(x \right)}} + \log{\left(\sqrt{\sin^{2}{\left(x \right)} + 1} + \sin{\left(x \right)} \right)}

  2. Simplificamos:

    xcos⁡(x)sin⁡2(x)+1+log⁡(sin⁡2(x)+1+sin⁡(x))\frac{x \cos{\left(x \right)}}{\sqrt{\sin^{2}{\left(x \right)} + 1}} + \log{\left(\sqrt{\sin^{2}{\left(x \right)} + 1} + \sin{\left(x \right)} \right)}

Respuesta:

xcos⁡(x)sin⁡2(x)+1+log⁡(sin⁡2(x)+1+sin⁡(x))\frac{x \cos{\left(x \right)}}{\sqrt{\sin^{2}{\left(x \right)} + 1}} + \log{\left(\sqrt{\sin^{2}{\left(x \right)} + 1} + \sin{\left(x \right)} \right)}

Gráfica
02468-8-6-4-2-1010-2020
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
  / cos(x)*sin(x)           \                                 
x*|---------------- + cos(x)|                                 
  |   _____________         |                                 
  |  /        2             |      /            _____________\
  \\/  1 + sin (x)          /      |           /        2    |
----------------------------- + log\sin(x) + \/  1 + sin (x) /
              _____________                                   
             /        2                                       
  sin(x) + \/  1 + sin (x)
x(cos⁡(x)+sin⁡(x)cos⁡(x)sin⁡2(x)+1)sin⁡2(x)+1+sin⁡(x)+log⁡(sin⁡2(x)+1+sin⁡(x))\frac{x \left(\cos{\left(x \right)} + \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\sqrt{\sin^{2}{\left(x \right)} + 1}}\right)}{\sqrt{\sin^{2}{\left(x \right)} + 1} + \sin{\left(x \right)}} + \log{\left(\sqrt{\sin^{2}{\left(x \right)} + 1} + \sin{\left(x \right)} \right)}
Simplificar
Segunda derivada [src] 
    /                                                                               2                 \                                  
    |                                                         /         sin(x)     \     2            |                                  
    |                                                         |1 + ----------------| *cos (x)         |                                  
    |                                                         |       _____________|                  |                                  
    |       2                  2              2       2       |      /        2    |                  |                                  
    |    sin (x)            cos (x)        cos (x)*sin (x)    \    \/  1 + sin (x) /                  |     /         sin(x)     \       
- x*|---------------- - ---------------- + ---------------- + ------------------------------- + sin(x)| + 2*|1 + ----------------|*cos(x)
    |   _____________      _____________                3/2         _____________                     |     |       _____________|       
    |  /        2         /        2       /       2   \           /        2                         |     |      /        2    |       
    \\/  1 + sin (x)    \/  1 + sin (x)    \1 + sin (x)/         \/  1 + sin (x)  + sin(x)            /     \    \/  1 + sin (x) /       
-----------------------------------------------------------------------------------------------------------------------------------------
                                                           _____________                                                                 
                                                          /        2                                                                     
                                                        \/  1 + sin (x)  + sin(x)
−x((1+sin⁡(x)sin⁡2(x)+1)2cos⁡2(x)sin⁡2(x)+1+sin⁡(x)+sin⁡(x)+sin⁡2(x)sin⁡2(x)+1−cos⁡2(x)sin⁡2(x)+1+sin⁡2(x)cos⁡2(x)(sin⁡2(x)+1)32)+2(1+sin⁡(x)sin⁡2(x)+1)cos⁡(x)sin⁡2(x)+1+sin⁡(x)\frac{- x \left(\frac{\left(1 + \frac{\sin{\left(x \right)}}{\sqrt{\sin^{2}{\left(x \right)} + 1}}\right)^{2} \cos^{2}{\left(x \right)}}{\sqrt{\sin^{2}{\left(x \right)} + 1} + \sin{\left(x \right)}} + \sin{\left(x \right)} + \frac{\sin^{2}{\left(x \right)}}{\sqrt{\sin^{2}{\left(x \right)} + 1}} - \frac{\cos^{2}{\left(x \right)}}{\sqrt{\sin^{2}{\left(x \right)} + 1}} + \frac{\sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{\frac{3}{2}}}\right) + 2 \left(1 + \frac{\sin{\left(x \right)}}{\sqrt{\sin^{2}{\left(x \right)} + 1}}\right) \cos{\left(x \right)}}{\sqrt{\sin^{2}{\left(x \right)} + 1} + \sin{\left(x \right)}}
Simplificar
Tercera derivada [src] 
                                                    /                                                                                      3                                                        /       2                  2              2       2             \\                                                      2        
                                                    |                                                                /         sin(x)     \     2                            /         sin(x)     \ |    sin (x)            cos (x)        cos (x)*sin (x)          ||                                /         sin(x)     \     2   
                                                    |                                                              2*|1 + ----------------| *cos (x)                       3*|1 + ----------------|*|---------------- - ---------------- + ---------------- + sin(x)||                              3*|1 + ----------------| *cos (x)
                                                    |                                                                |       _____________|                                  |       _____________| |   _____________      _____________                3/2         ||                                |       _____________|         
                    2                  2            |                                3               2               |      /        2    |                 2       3        |      /        2    | |  /        2         /        2       /       2   \            ||               2       2        |      /        2    |         
               3*sin (x)          3*cos (x)         |         4*sin(x)          3*sin (x)       3*cos (x)*sin(x)     \    \/  1 + sin (x) /            3*cos (x)*sin (x)     \    \/  1 + sin (x) / \\/  1 + sin (x)    \/  1 + sin (x)    \1 + sin (x)/            /|          3*cos (x)*sin (x)     \    \/  1 + sin (x) /         
-3*sin(x) - ---------------- + ---------------- + x*|-1 - ---------------- + ---------------- - ---------------- + --------------------------------- + ----------------- + ------------------------------------------------------------------------------------------|*cos(x) - ----------------- - ---------------------------------
               _____________      _____________     |        _____________                3/2                3/2                                 2                   5/2                                      _____________                                          |                        3/2          _____________             
              /        2         /        2         |       /        2       /       2   \      /       2   \         /   _____________         \       /       2   \                                        /        2                                              |           /       2   \            /        2                 
            \/  1 + sin (x)    \/  1 + sin (x)      |     \/  1 + sin (x)    \1 + sin (x)/      \1 + sin (x)/         |  /        2             |       \1 + sin (x)/                                      \/  1 + sin (x)  + sin(x)                                 |           \1 + sin (x)/          \/  1 + sin (x)  + sin(x)    
                                                    \                                                                 \\/  1 + sin (x)  + sin(x)/                                                                                                                    /                                                               
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                                                                                                                                                        /        2                                                                                                                                                                   
                                                                                                                                                      \/  1 + sin (x)  + sin(x)
x(2(1+sin⁡(x)sin⁡2(x)+1)3cos⁡2(x)(sin⁡2(x)+1+sin⁡(x))2+3(1+sin⁡(x)sin⁡2(x)+1)(sin⁡(x)+sin⁡2(x)sin⁡2(x)+1−cos⁡2(x)sin⁡2(x)+1+sin⁡2(x)cos⁡2(x)(sin⁡2(x)+1)32)sin⁡2(x)+1+sin⁡(x)−1−4sin⁡(x)sin⁡2(x)+1+3sin⁡3(x)(sin⁡2(x)+1)32−3sin⁡(x)cos⁡2(x)(sin⁡2(x)+1)32+3sin⁡3(x)cos⁡2(x)(sin⁡2(x)+1)52)cos⁡(x)−3(1+sin⁡(x)sin⁡2(x)+1)2cos⁡2(x)sin⁡2(x)+1+sin⁡(x)−3sin⁡(x)−3sin⁡2(x)sin⁡2(x)+1+3cos⁡2(x)sin⁡2(x)+1−3sin⁡2(x)cos⁡2(x)(sin⁡2(x)+1)32sin⁡2(x)+1+sin⁡(x)\frac{x \left(\frac{2 \left(1 + \frac{\sin{\left(x \right)}}{\sqrt{\sin^{2}{\left(x \right)} + 1}}\right)^{3} \cos^{2}{\left(x \right)}}{\left(\sqrt{\sin^{2}{\left(x \right)} + 1} + \sin{\left(x \right)}\right)^{2}} + \frac{3 \left(1 + \frac{\sin{\left(x \right)}}{\sqrt{\sin^{2}{\left(x \right)} + 1}}\right) \left(\sin{\left(x \right)} + \frac{\sin^{2}{\left(x \right)}}{\sqrt{\sin^{2}{\left(x \right)} + 1}} - \frac{\cos^{2}{\left(x \right)}}{\sqrt{\sin^{2}{\left(x \right)} + 1}} + \frac{\sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{\frac{3}{2}}}\right)}{\sqrt{\sin^{2}{\left(x \right)} + 1} + \sin{\left(x \right)}} - 1 - \frac{4 \sin{\left(x \right)}}{\sqrt{\sin^{2}{\left(x \right)} + 1}} + \frac{3 \sin^{3}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{\frac{3}{2}}} - \frac{3 \sin{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{\frac{3}{2}}} + \frac{3 \sin^{3}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{\frac{5}{2}}}\right) \cos{\left(x \right)} - \frac{3 \left(1 + \frac{\sin{\left(x \right)}}{\sqrt{\sin^{2}{\left(x \right)} + 1}}\right)^{2} \cos^{2}{\left(x \right)}}{\sqrt{\sin^{2}{\left(x \right)} + 1} + \sin{\left(x \right)}} - 3 \sin{\left(x \right)} - \frac{3 \sin^{2}{\left(x \right)}}{\sqrt{\sin^{2}{\left(x \right)} + 1}} + \frac{3 \cos^{2}{\left(x \right)}}{\sqrt{\sin^{2}{\left(x \right)} + 1}} - \frac{3 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{\frac{3}{2}}}}{\sqrt{\sin^{2}{\left(x \right)} + 1} + \sin{\left(x \right)}}
Simplificar
Gráfico
Derivada de xln(sinx+sqrt(1+(sinx)^2))
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