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x^sinx*lnx
Derivada de la función/ x^sin/ x*lnx/ x^sinx*lnx

Derivada de x^sinx*lnx

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

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Solución

Ha introducido [src] 
 sin(x)       
x      *log(x)
xsin⁡(x)log⁡(x)x^{\sin{\left(x \right)}} \log{\left(x \right)} 
x^sin(x)*log(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. No logro encontrar los pasos en la búsqueda de esta derivada.

      Perola derivada

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

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

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

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

  2. Simplificamos:

    x(log⁡(sin⁡(x))+1)log⁡(x)sin⁡sin⁡(x)(x)+xsin⁡(x)x\frac{x \left(\log{\left(\sin{\left(x \right)} \right)} + 1\right) \log{\left(x \right)} \sin^{\sin{\left(x \right)}}{\left(x \right)} + x^{\sin{\left(x \right)}}}{x}

Respuesta:

x(log⁡(sin⁡(x))+1)log⁡(x)sin⁡sin⁡(x)(x)+xsin⁡(x)x\frac{x \left(\log{\left(\sin{\left(x \right)} \right)} + 1\right) \log{\left(x \right)} \sin^{\sin{\left(x \right)}}{\left(x \right)} + x^{\sin{\left(x \right)}}}{x}

Gráfica
02468-8-6-4-2-1010-5050
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Trazar el gráfico f'(x)
Primera derivada [src] 
 sin(x)                                          
x          sin(x) /sin(x)                \       
------- + x      *|------ + cos(x)*log(x)|*log(x)
   x              \  x                   /
xsin⁡(x)(log⁡(x)cos⁡(x)+sin⁡(x)x)log⁡(x)+xsin⁡(x)xx^{\sin{\left(x \right)}} \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right) \log{\left(x \right)} + \frac{x^{\sin{\left(x \right)}}}{x}
Simplificar
Segunda derivada [src] 
        /                                                                                  /sin(x)                \\
        |       /                        2                                    \          2*|------ + cos(x)*log(x)||
 sin(x) |  1    |/sin(x)                \    sin(x)                   2*cos(x)|            \  x                   /|
x      *|- -- + ||------ + cos(x)*log(x)|  - ------ - log(x)*sin(x) + --------|*log(x) + --------------------------|
        |   2   |\  x                   /       2                        x    |                      x             |
        \  x    \                              x                              /                                    /
xsin⁡(x)(((log⁡(x)cos⁡(x)+sin⁡(x)x)2−log⁡(x)sin⁡(x)+2cos⁡(x)x−sin⁡(x)x2)log⁡(x)+2(log⁡(x)cos⁡(x)+sin⁡(x)x)x−1x2)x^{\sin{\left(x \right)}} \left(\left(\left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right)^{2} - \log{\left(x \right)} \sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x} - \frac{\sin{\left(x \right)}}{x^{2}}\right) \log{\left(x \right)} + \frac{2 \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right)}{x} - \frac{1}{x^{2}}\right)
Simplificar
Tercera derivada [src] 
        /                                                                                                                                                                                             /                        2                                    \\
        |                                                                                                                                                                                             |/sin(x)                \    sin(x)                   2*cos(x)||
        |                                                                                                                                                                /sin(x)                \   3*||------ + cos(x)*log(x)|  - ------ - log(x)*sin(x) + --------||
        |     /                          3                                                                                                                  \          3*|------ + cos(x)*log(x)|     |\  x                   /       2                        x    ||
 sin(x) |2    |  /sin(x)                \                    2*sin(x)   3*sin(x)   3*cos(x)     /sin(x)                \ /sin(x)                   2*cos(x)\|            \  x                   /     \                              x                              /|
x      *|-- - |- |------ + cos(x)*log(x)|  + cos(x)*log(x) - -------- + -------- + -------- + 3*|------ + cos(x)*log(x)|*|------ + log(x)*sin(x) - --------||*log(x) - -------------------------- + -----------------------------------------------------------------|
        | 3   |  \  x                   /                        3         x           2        \  x                   / |   2                        x    ||                       2                                               x                                |
        \x    \                                                 x                     x                                  \  x                              //                      x                                                                                 /
xsin⁡(x)(−(−(log⁡(x)cos⁡(x)+sin⁡(x)x)3+3(log⁡(x)cos⁡(x)+sin⁡(x)x)(log⁡(x)sin⁡(x)−2cos⁡(x)x+sin⁡(x)x2)+log⁡(x)cos⁡(x)+3sin⁡(x)x+3cos⁡(x)x2−2sin⁡(x)x3)log⁡(x)+3((log⁡(x)cos⁡(x)+sin⁡(x)x)2−log⁡(x)sin⁡(x)+2cos⁡(x)x−sin⁡(x)x2)x−3(log⁡(x)cos⁡(x)+sin⁡(x)x)x2+2x3)x^{\sin{\left(x \right)}} \left(- \left(- \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right)^{3} + 3 \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right) \left(\log{\left(x \right)} \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x} + \frac{\sin{\left(x \right)}}{x^{2}}\right) + \log{\left(x \right)} \cos{\left(x \right)} + \frac{3 \sin{\left(x \right)}}{x} + \frac{3 \cos{\left(x \right)}}{x^{2}} - \frac{2 \sin{\left(x \right)}}{x^{3}}\right) \log{\left(x \right)} + \frac{3 \left(\left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right)^{2} - \log{\left(x \right)} \sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x} - \frac{\sin{\left(x \right)}}{x^{2}}\right)}{x} - \frac{3 \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right)}{x^{2}} + \frac{2}{x^{3}}\right)
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Gráfico
Derivada de x^sinx*lnx
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