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x^sinc×sinx
Derivada de la función/ x^sin/ x^sinc×sinx

Derivada de x^sinc×sinx

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

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Solución

Ha introducido [src] 
 sinc(x)       
x       *sin(x)
xsinc⁡(x)sin⁡(x)x^{\operatorname{sinc}{\left(x \right)}} \sin{\left(x \right)} 
x^sinc(x)*sin(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)=xsinc⁡(x)f{\left(x \right)} = x^{\operatorname{sinc}{\left(x \right)}}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. No logro encontrar los pasos en la búsqueda de esta derivada.

      Perola derivada

      (log⁡(sinc⁡(x))+1)sinc⁡sinc⁡(x)(x)\left(\log{\left(\operatorname{sinc}{\left(x \right)} \right)} + 1\right) \operatorname{sinc}^{\operatorname{sinc}{\left(x \right)}}{\left(x \right)}

    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: xsinc⁡(x)cos⁡(x)+(log⁡(sinc⁡(x))+1)sin⁡(x)sinc⁡sinc⁡(x)(x)x^{\operatorname{sinc}{\left(x \right)}} \cos{\left(x \right)} + \left(\log{\left(\operatorname{sinc}{\left(x \right)} \right)} + 1\right) \sin{\left(x \right)} \operatorname{sinc}^{\operatorname{sinc}{\left(x \right)}}{\left(x \right)}

Respuesta:

xsinc⁡(x)cos⁡(x)+(log⁡(sinc⁡(x))+1)sin⁡(x)sinc⁡sinc⁡(x)(x)x^{\operatorname{sinc}{\left(x \right)}} \cos{\left(x \right)} + \left(\log{\left(\operatorname{sinc}{\left(x \right)} \right)} + 1\right) \sin{\left(x \right)} \operatorname{sinc}^{\operatorname{sinc}{\left(x \right)}}{\left(x \right)}

Gráfica
02468-8-6-4-2-10102.5-2.5
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Primera derivada [src] 
                           /          //-sin(x) + x*cos(x)            \       \       
                           |          ||------------------  for x != 0|       |       
 sinc(x)           sinc(x) |sinc(x)   ||         2                    |       |       
x       *cos(x) + x       *|------- + |<        x                     |*log(x)|*sin(x)
                           |   x      ||                              |       |       
                           |          ||        0           otherwise |       |       
                           \          \\                              /       /
xsinc⁡(x)(({xcos⁡(x)−sin⁡(x)x2for x≠00otherwise)log⁡(x)+sinc⁡(x)x)sin⁡(x)+xsinc⁡(x)cos⁡(x)x^{\operatorname{sinc}{\left(x \right)}} \left(\left(\begin{cases} \frac{x \cos{\left(x \right)} - \sin{\left(x \right)}}{x^{2}} & \text{for}\: x \neq 0 \\0 & \text{otherwise} \end{cases}\right) \log{\left(x \right)} + \frac{\operatorname{sinc}{\left(x \right)}}{x}\right) \sin{\left(x \right)} + x^{\operatorname{sinc}{\left(x \right)}} \cos{\left(x \right)}
Simplificar
Segunda derivada [src] 
         /          /                                                                                                                                //-sin(x) + x*cos(x)            \\                                                                       \
         |          |                                                                                                                                ||------------------  for x != 0||                                                                       |
         |          |                                                                                                                                ||         2                    ||                                                                       |
         |          |                                                    2   // /2*(-sin(x) + x*cos(x))         \             \                    2*|<        x                     ||                                                                       |
         |          |/          //-sin(x) + x*cos(x)            \       \    ||-|---------------------- + sin(x)|             |                      ||                              ||            /          //-sin(x) + x*cos(x)            \       \       |
         |          ||          ||------------------  for x != 0|       |    || |           2                   |             |                      ||        0           otherwise ||            |          ||------------------  for x != 0|       |       |
 sinc(x) |          ||sinc(x)   ||         2                    |       |    || \          x                    /             |          sinc(x)     \\                              /|            |sinc(x)   ||         2                    |       |       |
x       *|-sin(x) + ||------- + |<        x                     |*log(x)|  + |<-----------------------------------  for x != 0|*log(x) - ------- + -----------------------------------|*sin(x) + 2*|------- + |<        x                     |*log(x)|*cos(x)|
         |          ||   x      ||                              |       |    ||                 x                             |              2                      x                 |            |   x      ||                              |       |       |
         |          ||          ||        0           otherwise |       |    ||                                               |             x                                         |            |          ||        0           otherwise |       |       |
         |          |\          \\                              /       /    ||                 0                   otherwise |                                                       |            \          \\                              /       /       |
         \          \                                                        \\                                               /                                                       /                                                                       /
xsinc⁡(x)(2(({xcos⁡(x)−sin⁡(x)x2for x≠00otherwise)log⁡(x)+sinc⁡(x)x)cos⁡(x)+((({xcos⁡(x)−sin⁡(x)x2for x≠00otherwise)log⁡(x)+sinc⁡(x)x)2+({−sin⁡(x)+2(xcos⁡(x)−sin⁡(x))x2xfor x≠00otherwise)log⁡(x)+2({xcos⁡(x)−sin⁡(x)x2for x≠00otherwise)x−sinc⁡(x)x2)sin⁡(x)−sin⁡(x))x^{\operatorname{sinc}{\left(x \right)}} \left(2 \left(\left(\begin{cases} \frac{x \cos{\left(x \right)} - \sin{\left(x \right)}}{x^{2}} & \text{for}\: x \neq 0 \\0 & \text{otherwise} \end{cases}\right) \log{\left(x \right)} + \frac{\operatorname{sinc}{\left(x \right)}}{x}\right) \cos{\left(x \right)} + \left(\left(\left(\begin{cases} \frac{x \cos{\left(x \right)} - \sin{\left(x \right)}}{x^{2}} & \text{for}\: x \neq 0 \\0 & \text{otherwise} \end{cases}\right) \log{\left(x \right)} + \frac{\operatorname{sinc}{\left(x \right)}}{x}\right)^{2} + \left(\begin{cases} - \frac{\sin{\left(x \right)} + \frac{2 \left(x \cos{\left(x \right)} - \sin{\left(x \right)}\right)}{x^{2}}}{x} & \text{for}\: x \neq 0 \\0 & \text{otherwise} \end{cases}\right) \log{\left(x \right)} + \frac{2 \left(\begin{cases} \frac{x \cos{\left(x \right)} - \sin{\left(x \right)}}{x^{2}} & \text{for}\: x \neq 0 \\0 & \text{otherwise} \end{cases}\right)}{x} - \frac{\operatorname{sinc}{\left(x \right)}}{x^{2}}\right) \sin{\left(x \right)} - \sin{\left(x \right)}\right)
Simplificar
Tercera derivada [src] 
         /          /                                                                                                                                                                                // /2*(-sin(x) + x*cos(x))         \             \                                                                                                                                                                     \                                                                                                                                                                                                                                                      \
         |          |                                                                                                                                                                                ||-|---------------------- + sin(x)|             |                                                                                                                                                                     |                                                                                                                                                                                                                                                      |
         |          |                                                                                                                              //-sin(x) + x*cos(x)            \                 || |           2                   |             |                                                          /                                                                        //-sin(x) + x*cos(x)            \\|                                                                            /                                                                                                                                //-sin(x) + x*cos(x)            \\       |
         |          |                                                                                                                              ||------------------  for x != 0|                 || \          x                    /             |                                                          |                                                                        ||------------------  for x != 0|||                                                                            |                                                                                                                                ||------------------  for x != 0||       |
         |          |                                                                                                                              ||         2                    |               3*|<-----------------------------------  for x != 0|                                                          |                                                                        ||         2                    |||                                                                            |                                                                                                                                ||         2                    ||       |
         |          |                                                    3   //          3*sin(x)   6*(-sin(x) + x*cos(x))            \          3*|<        x                     |                 ||                 x                             |                                                          |// /2*(-sin(x) + x*cos(x))         \             \                    2*|<        x                     |||                                                                            |                                                    2   // /2*(-sin(x) + x*cos(x))         \             \                    2*|<        x                     ||       |
         |          |/          //-sin(x) + x*cos(x)            \       \    ||-cos(x) + -------- + ----------------------            |            ||                              |                 ||                                               |     /          //-sin(x) + x*cos(x)            \       \ |||-|---------------------- + sin(x)|             |                      ||                              |||            /          //-sin(x) + x*cos(x)            \       \            |/          //-sin(x) + x*cos(x)            \       \    ||-|---------------------- + sin(x)|             |                      ||                              ||       |
         |          ||          ||------------------  for x != 0|       |    ||             x                  3                      |            ||        0           otherwise |                 ||                 0                   otherwise |     |          ||------------------  for x != 0|       | ||| |           2                   |             |                      ||        0           otherwise |||            |          ||------------------  for x != 0|       |            ||          ||------------------  for x != 0|       |    || |           2                   |             |                      ||        0           otherwise ||       |
 sinc(x) |          ||sinc(x)   ||         2                    |       |    ||                               x                       |            \\                              /   2*sinc(x)     \\                                               /     |sinc(x)   ||         2                    |       | ||| \          x                    /             |          sinc(x)     \\                              /||            |sinc(x)   ||         2                    |       |            ||sinc(x)   ||         2                    |       |    || \          x                    /             |          sinc(x)     \\                              /|       |
x       *|-cos(x) + ||------- + |<        x                     |*log(x)|  + |<-------------------------------------------  for x != 0|*log(x) - ----------------------------------- + --------- + ---------------------------------------------------- + 3*|------- + |<        x                     |*log(x)|*||<-----------------------------------  for x != 0|*log(x) - ------- + -----------------------------------||*sin(x) - 3*|------- + |<        x                     |*log(x)|*sin(x) + 3*||------- + |<        x                     |*log(x)|  + |<-----------------------------------  for x != 0|*log(x) - ------- + -----------------------------------|*cos(x)|
         |          ||   x      ||                              |       |    ||                     x                                 |                            2                        3                               x                               |   x      ||                              |       | |||                 x                             |              2                      x                 ||            |   x      ||                              |       |            ||   x      ||                              |       |    ||                 x                             |              2                      x                 |       |
         |          ||          ||        0           otherwise |       |    ||                                                       |                           x                        x                                                                |          ||        0           otherwise |       | |||                                               |             x                                         ||            |          ||        0           otherwise |       |            ||          ||        0           otherwise |       |    ||                                               |             x                                         |       |
         |          |\          \\                              /       /    ||                     0                       otherwise |                                                                                                                     \          \\                              /       / |||                 0                   otherwise |                                                       ||            \          \\                              /       /            |\          \\                              /       /    ||                 0                   otherwise |                                                       |       |
         \          \                                                        \\                                                       /                                                                                                                                                                          \\\                                               /                                                       //                                                                            \                                                        \\                                               /                                                       /       /
xsinc⁡(x)(−3(({xcos⁡(x)−sin⁡(x)x2for x≠00otherwise)log⁡(x)+sinc⁡(x)x)sin⁡(x)+3((({xcos⁡(x)−sin⁡(x)x2for x≠00otherwise)log⁡(x)+sinc⁡(x)x)2+({−sin⁡(x)+2(xcos⁡(x)−sin⁡(x))x2xfor x≠00otherwise)log⁡(x)+2({xcos⁡(x)−sin⁡(x)x2for x≠00otherwise)x−sinc⁡(x)x2)cos⁡(x)+((({xcos⁡(x)−sin⁡(x)x2for x≠00otherwise)log⁡(x)+sinc⁡(x)x)3+3(({xcos⁡(x)−sin⁡(x)x2for x≠00otherwise)log⁡(x)+sinc⁡(x)x)(({−sin⁡(x)+2(xcos⁡(x)−sin⁡(x))x2xfor x≠00otherwise)log⁡(x)+2({xcos⁡(x)−sin⁡(x)x2for x≠00otherwise)x−sinc⁡(x)x2)+({−cos⁡(x)+3sin⁡(x)x+6(xcos⁡(x)−sin⁡(x))x3xfor x≠00otherwise)log⁡(x)+3({−sin⁡(x)+2(xcos⁡(x)−sin⁡(x))x2xfor x≠00otherwise)x−3({xcos⁡(x)−sin⁡(x)x2for x≠00otherwise)x2+2sinc⁡(x)x3)sin⁡(x)−cos⁡(x))x^{\operatorname{sinc}{\left(x \right)}} \left(- 3 \left(\left(\begin{cases} \frac{x \cos{\left(x \right)} - \sin{\left(x \right)}}{x^{2}} & \text{for}\: x \neq 0 \\0 & \text{otherwise} \end{cases}\right) \log{\left(x \right)} + \frac{\operatorname{sinc}{\left(x \right)}}{x}\right) \sin{\left(x \right)} + 3 \left(\left(\left(\begin{cases} \frac{x \cos{\left(x \right)} - \sin{\left(x \right)}}{x^{2}} & \text{for}\: x \neq 0 \\0 & \text{otherwise} \end{cases}\right) \log{\left(x \right)} + \frac{\operatorname{sinc}{\left(x \right)}}{x}\right)^{2} + \left(\begin{cases} - \frac{\sin{\left(x \right)} + \frac{2 \left(x \cos{\left(x \right)} - \sin{\left(x \right)}\right)}{x^{2}}}{x} & \text{for}\: x \neq 0 \\0 & \text{otherwise} \end{cases}\right) \log{\left(x \right)} + \frac{2 \left(\begin{cases} \frac{x \cos{\left(x \right)} - \sin{\left(x \right)}}{x^{2}} & \text{for}\: x \neq 0 \\0 & \text{otherwise} \end{cases}\right)}{x} - \frac{\operatorname{sinc}{\left(x \right)}}{x^{2}}\right) \cos{\left(x \right)} + \left(\left(\left(\begin{cases} \frac{x \cos{\left(x \right)} - \sin{\left(x \right)}}{x^{2}} & \text{for}\: x \neq 0 \\0 & \text{otherwise} \end{cases}\right) \log{\left(x \right)} + \frac{\operatorname{sinc}{\left(x \right)}}{x}\right)^{3} + 3 \left(\left(\begin{cases} \frac{x \cos{\left(x \right)} - \sin{\left(x \right)}}{x^{2}} & \text{for}\: x \neq 0 \\0 & \text{otherwise} \end{cases}\right) \log{\left(x \right)} + \frac{\operatorname{sinc}{\left(x \right)}}{x}\right) \left(\left(\begin{cases} - \frac{\sin{\left(x \right)} + \frac{2 \left(x \cos{\left(x \right)} - \sin{\left(x \right)}\right)}{x^{2}}}{x} & \text{for}\: x \neq 0 \\0 & \text{otherwise} \end{cases}\right) \log{\left(x \right)} + \frac{2 \left(\begin{cases} \frac{x \cos{\left(x \right)} - \sin{\left(x \right)}}{x^{2}} & \text{for}\: x \neq 0 \\0 & \text{otherwise} \end{cases}\right)}{x} - \frac{\operatorname{sinc}{\left(x \right)}}{x^{2}}\right) + \left(\begin{cases} \frac{- \cos{\left(x \right)} + \frac{3 \sin{\left(x \right)}}{x} + \frac{6 \left(x \cos{\left(x \right)} - \sin{\left(x \right)}\right)}{x^{3}}}{x} & \text{for}\: x \neq 0 \\0 & \text{otherwise} \end{cases}\right) \log{\left(x \right)} + \frac{3 \left(\begin{cases} - \frac{\sin{\left(x \right)} + \frac{2 \left(x \cos{\left(x \right)} - \sin{\left(x \right)}\right)}{x^{2}}}{x} & \text{for}\: x \neq 0 \\0 & \text{otherwise} \end{cases}\right)}{x} - \frac{3 \left(\begin{cases} \frac{x \cos{\left(x \right)} - \sin{\left(x \right)}}{x^{2}} & \text{for}\: x \neq 0 \\0 & \text{otherwise} \end{cases}\right)}{x^{2}} + \frac{2 \operatorname{sinc}{\left(x \right)}}{x^{3}}\right) \sin{\left(x \right)} - \cos{\left(x \right)}\right)
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Gráfico
Derivada de x^sinc×sinx
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