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Derivada de la función/ x*sin/ sin(x)/ y=(x*sin(x))^(8lnsin(x))

Derivada de y=(x*sin(x))^(8lnsin(x))

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

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
          8*log(sin(x))
(x*sin(x))
(xsin⁡(x))8log⁡(sin⁡(x))\left(x \sin{\left(x \right)}\right)^{8 \log{\left(\sin{\left(x \right)} \right)}} 
(x*sin(x))^(8*log(sin(x)))
Solución detallada

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

    Perola derivada

    (8log⁡(sin⁡(x)))8log⁡(sin⁡(x))(log⁡(8log⁡(sin⁡(x)))+1)\left(8 \log{\left(\sin{\left(x \right)} \right)}\right)^{8 \log{\left(\sin{\left(x \right)} \right)}} \left(\log{\left(8 \log{\left(\sin{\left(x \right)} \right)} \right)} + 1\right)

Respuesta:

(8log⁡(sin⁡(x)))8log⁡(sin⁡(x))(log⁡(8log⁡(sin⁡(x)))+1)\left(8 \log{\left(\sin{\left(x \right)} \right)}\right)^{8 \log{\left(\sin{\left(x \right)} \right)}} \left(\log{\left(8 \log{\left(\sin{\left(x \right)} \right)} \right)} + 1\right)

Gráfica
02468-8-6-4-2-1010-5e585e58
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
          8*log(sin(x)) /8*cos(x)*log(x*sin(x))   8*(x*cos(x) + sin(x))*log(sin(x))\
(x*sin(x))             *|---------------------- + ---------------------------------|
                        \        sin(x)                        x*sin(x)            /
(xsin⁡(x))8log⁡(sin⁡(x))(8log⁡(xsin⁡(x))cos⁡(x)sin⁡(x)+8(xcos⁡(x)+sin⁡(x))log⁡(sin⁡(x))xsin⁡(x))\left(x \sin{\left(x \right)}\right)^{8 \log{\left(\sin{\left(x \right)} \right)}} \left(\frac{8 \log{\left(x \sin{\left(x \right)} \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{8 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(\sin{\left(x \right)} \right)}}{x \sin{\left(x \right)}}\right)
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Segunda derivada [src] 
                          /                                                                           2                                                                                                                                                                       \
                          |                   /                       (x*cos(x) + sin(x))*log(sin(x))\                                                                                                                                                                        |
                          |                 8*|cos(x)*log(x*sin(x)) + -------------------------------|       2                                                                                                                                                                |
            8*log(sin(x)) |                   \                                      x               /    cos (x)*log(x*sin(x))   (-2*cos(x) + x*sin(x))*log(sin(x))   (x*cos(x) + sin(x))*log(sin(x))   2*(x*cos(x) + sin(x))*cos(x)   (x*cos(x) + sin(x))*cos(x)*log(sin(x))|
8*(x*sin(x))             *|-log(x*sin(x)) + ----------------------------------------------------------- - --------------------- - ---------------------------------- - ------------------------------- + ---------------------------- - --------------------------------------|
                          |                                              2                                          2                          x*sin(x)                            2                                   2                                   2                  |
                          \                                           sin (x)                                    sin (x)                                                          x *sin(x)                       x*sin (x)                           x*sin (x)               /
8(xsin⁡(x))8log⁡(sin⁡(x))(8(log⁡(xsin⁡(x))cos⁡(x)+(xcos⁡(x)+sin⁡(x))log⁡(sin⁡(x))x)2sin⁡2(x)−log⁡(xsin⁡(x))−log⁡(xsin⁡(x))cos⁡2(x)sin⁡2(x)−(xsin⁡(x)−2cos⁡(x))log⁡(sin⁡(x))xsin⁡(x)−(xcos⁡(x)+sin⁡(x))log⁡(sin⁡(x))cos⁡(x)xsin⁡2(x)+2(xcos⁡(x)+sin⁡(x))cos⁡(x)xsin⁡2(x)−(xcos⁡(x)+sin⁡(x))log⁡(sin⁡(x))x2sin⁡(x))8 \left(x \sin{\left(x \right)}\right)^{8 \log{\left(\sin{\left(x \right)} \right)}} \left(\frac{8 \left(\log{\left(x \sin{\left(x \right)} \right)} \cos{\left(x \right)} + \frac{\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(\sin{\left(x \right)} \right)}}{x}\right)^{2}}{\sin^{2}{\left(x \right)}} - \log{\left(x \sin{\left(x \right)} \right)} - \frac{\log{\left(x \sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{\left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \log{\left(\sin{\left(x \right)} \right)}}{x \sin{\left(x \right)}} - \frac{\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}}{x \sin^{2}{\left(x \right)}} + \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \cos{\left(x \right)}}{x \sin^{2}{\left(x \right)}} - \frac{\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(\sin{\left(x \right)} \right)}}{x^{2} \sin{\left(x \right)}}\right)
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Tercera derivada [src] 
                          /                                                                                                                                                                                                                                                                                                                                                                   3                                                                                                                                                                                                                                                                                                                                                                                                                 \
                          |                                                                                                                                                                                                                                                                                                           /                       (x*cos(x) + sin(x))*log(sin(x))\                                                                                                                                                                                                                                                                                                                                                                                                                  |
                          |                                                              /   2                                                                                                                                                                                \                                                    64*|cos(x)*log(x*sin(x)) + -------------------------------|                                                                                                                                                          3                         2                                                                                                2                                                                                                                            |
            8*log(sin(x)) |     /                       (x*cos(x) + sin(x))*log(sin(x))\ |cos (x)*log(x*sin(x))   (-2*cos(x) + x*sin(x))*log(sin(x))   (x*cos(x) + sin(x))*log(sin(x))   2*(x*cos(x) + sin(x))*cos(x)   (x*cos(x) + sin(x))*cos(x)*log(sin(x))                |   3*(x*cos(x) + sin(x))                               \                                      x               /    (x*cos(x) + sin(x))*log(sin(x))   (3*sin(x) + x*cos(x))*log(sin(x))   2*(x*cos(x) + sin(x))*log(sin(x))   2*(-2*cos(x) + x*sin(x))*log(sin(x))   2*cos (x)*log(x*sin(x))   6*cos (x)*(x*cos(x) + sin(x))   3*(-2*cos(x) + x*sin(x))*cos(x)   3*(x*cos(x) + sin(x))*cos(x)   2*cos (x)*(x*cos(x) + sin(x))*log(sin(x))   2*(-2*cos(x) + x*sin(x))*cos(x)*log(sin(x))   2*(x*cos(x) + sin(x))*cos(x)*log(sin(x))|
8*(x*sin(x))             *|- 24*|cos(x)*log(x*sin(x)) + -------------------------------|*|--------------------- + ---------------------------------- + ------------------------------- - ---------------------------- + -------------------------------------- + log(x*sin(x))| - --------------------- + 2*cos(x)*log(x*sin(x)) + ------------------------------------------------------------ + ------------------------------- - --------------------------------- + --------------------------------- + ------------------------------------ + ----------------------- - ----------------------------- - ------------------------------- - ---------------------------- + ----------------------------------------- + ------------------------------------------- + ----------------------------------------|
                          |     \                                      x               / |          2                          x*sin(x)                            2                                   2                                   2                                  |             x                                                                   2                                                x                                  x                                    3                                    2                               2                             2                            x*sin(x)                        2                                         2                                        x*sin(x)                                    2                       |
                          \                                                              \       sin (x)                                                          x *sin(x)                       x*sin (x)                           x*sin (x)                               /                                                                              sin (x)                                                                                                                    x                                    x                             sin (x)                     x*sin (x)                                                        x *sin(x)                             x*sin (x)                                                                                x *sin(x)                /
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                                                                                                                                                                                                                                                                                                                                                                                                      sin(x)
8(xsin⁡(x))8log⁡(sin⁡(x))(64(log⁡(xsin⁡(x))cos⁡(x)+(xcos⁡(x)+sin⁡(x))log⁡(sin⁡(x))x)3sin⁡2(x)−24(log⁡(xsin⁡(x))cos⁡(x)+(xcos⁡(x)+sin⁡(x))log⁡(sin⁡(x))x)(log⁡(xsin⁡(x))+log⁡(xsin⁡(x))cos⁡2(x)sin⁡2(x)+(xsin⁡(x)−2cos⁡(x))log⁡(sin⁡(x))xsin⁡(x)+(xcos⁡(x)+sin⁡(x))log⁡(sin⁡(x))cos⁡(x)xsin⁡2(x)−2(xcos⁡(x)+sin⁡(x))cos⁡(x)xsin⁡2(x)+(xcos⁡(x)+sin⁡(x))log⁡(sin⁡(x))x2sin⁡(x))+2log⁡(xsin⁡(x))cos⁡(x)+2log⁡(xsin⁡(x))cos⁡3(x)sin⁡2(x)+2(xsin⁡(x)−2cos⁡(x))log⁡(sin⁡(x))cos⁡(x)xsin⁡(x)−3(xsin⁡(x)−2cos⁡(x))cos⁡(x)xsin⁡(x)+(xcos⁡(x)+sin⁡(x))log⁡(sin⁡(x))x+2(xcos⁡(x)+sin⁡(x))log⁡(sin⁡(x))cos⁡2(x)xsin⁡2(x)−3(xcos⁡(x)+sin⁡(x))x−6(xcos⁡(x)+sin⁡(x))cos⁡2(x)xsin⁡2(x)−(xcos⁡(x)+3sin⁡(x))log⁡(sin⁡(x))x+2(xsin⁡(x)−2cos⁡(x))log⁡(sin⁡(x))x2+2(xcos⁡(x)+sin⁡(x))log⁡(sin⁡(x))cos⁡(x)x2sin⁡(x)−3(xcos⁡(x)+sin⁡(x))cos⁡(x)x2sin⁡(x)+2(xcos⁡(x)+sin⁡(x))log⁡(sin⁡(x))x3)sin⁡(x)\frac{8 \left(x \sin{\left(x \right)}\right)^{8 \log{\left(\sin{\left(x \right)} \right)}} \left(\frac{64 \left(\log{\left(x \sin{\left(x \right)} \right)} \cos{\left(x \right)} + \frac{\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(\sin{\left(x \right)} \right)}}{x}\right)^{3}}{\sin^{2}{\left(x \right)}} - 24 \left(\log{\left(x \sin{\left(x \right)} \right)} \cos{\left(x \right)} + \frac{\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(\sin{\left(x \right)} \right)}}{x}\right) \left(\log{\left(x \sin{\left(x \right)} \right)} + \frac{\log{\left(x \sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{\left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \log{\left(\sin{\left(x \right)} \right)}}{x \sin{\left(x \right)}} + \frac{\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}}{x \sin^{2}{\left(x \right)}} - \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \cos{\left(x \right)}}{x \sin^{2}{\left(x \right)}} + \frac{\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(\sin{\left(x \right)} \right)}}{x^{2} \sin{\left(x \right)}}\right) + 2 \log{\left(x \sin{\left(x \right)} \right)} \cos{\left(x \right)} + \frac{2 \log{\left(x \sin{\left(x \right)} \right)} \cos^{3}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{2 \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \log{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}}{x \sin{\left(x \right)}} - \frac{3 \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \cos{\left(x \right)}}{x \sin{\left(x \right)}} + \frac{\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(\sin{\left(x \right)} \right)}}{x} + \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}}{x \sin^{2}{\left(x \right)}} - \frac{3 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)}{x} - \frac{6 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \cos^{2}{\left(x \right)}}{x \sin^{2}{\left(x \right)}} - \frac{\left(x \cos{\left(x \right)} + 3 \sin{\left(x \right)}\right) \log{\left(\sin{\left(x \right)} \right)}}{x} + \frac{2 \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \log{\left(\sin{\left(x \right)} \right)}}{x^{2}} + \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}}{x^{2} \sin{\left(x \right)}} - \frac{3 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \cos{\left(x \right)}}{x^{2} \sin{\left(x \right)}} + \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(\sin{\left(x \right)} \right)}}{x^{3}}\right)}{\sin{\left(x \right)}}
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