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y=((cos^3)x)^(lnx)
Derivada de la función/ cos^3/ y=((cos^3)x)^(lnx)

Derivada de y=((cos^3)x)^(lnx)

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

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Solución

Ha introducido [src] 
           log(x)
/   3     \      
\cos (x)*x/
(xcos⁡3(x))log⁡(x)\left(x \cos^{3}{\left(x \right)}\right)^{\log{\left(x \right)}} 
(cos(x)^3*x)^log(x)
Solución detallada

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

    Perola derivada

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

Respuesta:

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

Gráfica
02468-8-6-4-2-1010-1000010000
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
           log(x) /   /   3     \   /   3             2          \       \
/   3     \       |log\cos (x)*x/   \cos (x) - 3*x*cos (x)*sin(x)/*log(x)|
\cos (x)*x/      *|-------------- + -------------------------------------|
                  |      x                             3                 |
                  \                               x*cos (x)              /
(xcos⁡3(x))log⁡(x)((−3xsin⁡(x)cos⁡2(x)+cos⁡3(x))log⁡(x)xcos⁡3(x)+log⁡(xcos⁡3(x))x)\left(x \cos^{3}{\left(x \right)}\right)^{\log{\left(x \right)}} \left(\frac{\left(- 3 x \sin{\left(x \right)} \cos^{2}{\left(x \right)} + \cos^{3}{\left(x \right)}\right) \log{\left(x \right)}}{x \cos^{3}{\left(x \right)}} + \frac{\log{\left(x \cos^{3}{\left(x \right)} \right)}}{x}\right)
Simplificar
Segunda derivada [src] 
                  /                                                  2                                                                                                                                                                            \
                  |/     /     3   \   (-cos(x) + 3*x*sin(x))*log(x)\                                                                                                                                                                             |
           log(x) ||- log\x*cos (x)/ + -----------------------------|       /     3   \     /     2             2                     \                                                                                                           |
/     3   \       |\                               cos(x)           /    log\x*cos (x)/   3*\x*cos (x) - 2*x*sin (x) + 2*cos(x)*sin(x)/*log(x)   2*(-cos(x) + 3*x*sin(x))   (-cos(x) + 3*x*sin(x))*log(x)   3*(-cos(x) + 3*x*sin(x))*log(x)*sin(x)|
\x*cos (x)/      *|--------------------------------------------------- - -------------- - ---------------------------------------------------- - ------------------------ + ----------------------------- - --------------------------------------|
                  |                         x                                  x                                   2                                     x*cos(x)                      x*cos(x)                               2                   |
                  \                                                                                             cos (x)                                                                                                    cos (x)                /
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                         x
(xcos⁡3(x))log⁡(x)(−3(3xsin⁡(x)−cos⁡(x))log⁡(x)sin⁡(x)cos⁡2(x)−3(−2xsin⁡2(x)+xcos⁡2(x)+2sin⁡(x)cos⁡(x))log⁡(x)cos⁡2(x)+(3xsin⁡(x)−cos⁡(x))log⁡(x)xcos⁡(x)−2(3xsin⁡(x)−cos⁡(x))xcos⁡(x)+((3xsin⁡(x)−cos⁡(x))log⁡(x)cos⁡(x)−log⁡(xcos⁡3(x)))2x−log⁡(xcos⁡3(x))x)x\frac{\left(x \cos^{3}{\left(x \right)}\right)^{\log{\left(x \right)}} \left(- \frac{3 \left(3 x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \log{\left(x \right)} \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{3 \left(- 2 x \sin^{2}{\left(x \right)} + x \cos^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)}\right) \log{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\left(3 x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \log{\left(x \right)}}{x \cos{\left(x \right)}} - \frac{2 \left(3 x \sin{\left(x \right)} - \cos{\left(x \right)}\right)}{x \cos{\left(x \right)}} + \frac{\left(\frac{\left(3 x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \log{\left(x \right)}}{\cos{\left(x \right)}} - \log{\left(x \cos^{3}{\left(x \right)} \right)}\right)^{2}}{x} - \frac{\log{\left(x \cos^{3}{\left(x \right)} \right)}}{x}\right)}{x}
Simplificar
Tercera derivada [src] 
                  /                                                                                                                                                                                                                                                                                               /   /     3   \                                /     2             2                     \                                                                                \                                                                                                                                                                                                                                                                                                         \
                  |                                                    3                                                                                                                                                                                       /     /     3   \   (-cos(x) + 3*x*sin(x))*log(x)\ |log\x*cos (x)/   2*(-cos(x) + 3*x*sin(x))   3*\x*cos (x) - 2*x*sin (x) + 2*cos(x)*sin(x)/*log(x)   (-cos(x) + 3*x*sin(x))*log(x)   3*(-cos(x) + 3*x*sin(x))*log(x)*sin(x)|                                                                                                                                                                                                                                                                                                         |
                  |  /     /     3   \   (-cos(x) + 3*x*sin(x))*log(x)\                                                                                                                                                                                      3*|- log\x*cos (x)/ + -----------------------------|*|-------------- + ------------------------ + ---------------------------------------------------- - ----------------------------- + --------------------------------------|                                                                                                                                                                                                                                                                                                         |
           log(x) |  |- log\x*cos (x)/ + -----------------------------|         /     3   \     /     2             2                     \                                       /     3           2                    3             2          \            \                               cos(x)           / |      x                  x*cos(x)                                    2                                        x*cos(x)                               2                   |                                 /     2             2                     \                       2                                                                                                          /     2             2                     \                                                |
/     3   \       |  \                               cos(x)           /    2*log\x*cos (x)/   9*\x*cos (x) - 2*x*sin (x) + 2*cos(x)*sin(x)/   3*(-cos(x) + 3*x*sin(x))*log(x)   3*\3*cos (x) - 6*sin (x)*cos(x) + 2*x*sin (x) - 7*x*cos (x)*sin(x)/*log(x)                                                        \                                                                  cos (x)                                                                         cos (x)                /   6*(-cos(x) + 3*x*sin(x))   18*\x*cos (x) - 2*x*sin (x) + 2*cos(x)*sin(x)/*log(x)*sin(x)   12*sin (x)*(-cos(x) + 3*x*sin(x))*log(x)   9*(-cos(x) + 3*x*sin(x))*sin(x)   2*(-cos(x) + 3*x*sin(x))*log(x)   6*\x*cos (x) - 2*x*sin (x) + 2*cos(x)*sin(x)/*log(x)   6*(-cos(x) + 3*x*sin(x))*log(x)*sin(x)|
\x*cos (x)/      *|- --------------------------------------------------- + ---------------- - --------------------------------------------- - ------------------------------- - -------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------ - ------------------------------------------------------------ - ---------------------------------------- - ------------------------------- - ------------------------------- + ---------------------------------------------------- + --------------------------------------|
                  |                            2                                   2                                 2                                     cos(x)                                                   3                                                                                                                                                       x                                                                                                                           2                                               3                                                    3                                       2                             2                                               2                                               2                  |
                  \                           x                                   x                             x*cos (x)                                                                                        cos (x)                                                                                                                                                                                                                                                                               x *cos(x)                                     cos (x)                                              cos (x)                               x*cos (x)                         x *cos(x)                                   x*cos (x)                                       x*cos (x)               /
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                                                                                                                                                                                                                                                                                                                                                                                                   x
(xcos⁡3(x))log⁡(x)(−12(3xsin⁡(x)−cos⁡(x))log⁡(x)sin⁡2(x)cos⁡3(x)−3(3xsin⁡(x)−cos⁡(x))log⁡(x)cos⁡(x)−18(−2xsin⁡2(x)+xcos⁡2(x)+2sin⁡(x)cos⁡(x))log⁡(x)sin⁡(x)cos⁡3(x)−3(2xsin⁡3(x)−7xsin⁡(x)cos⁡2(x)−6sin⁡2(x)cos⁡(x)+3cos⁡3(x))log⁡(x)cos⁡3(x)+6(3xsin⁡(x)−cos⁡(x))log⁡(x)sin⁡(x)xcos⁡2(x)−9(3xsin⁡(x)−cos⁡(x))sin⁡(x)xcos⁡2(x)+3((3xsin⁡(x)−cos⁡(x))log⁡(x)cos⁡(x)−log⁡(xcos⁡3(x)))(3(3xsin⁡(x)−cos⁡(x))log⁡(x)sin⁡(x)cos⁡2(x)+3(−2xsin⁡2(x)+xcos⁡2(x)+2sin⁡(x)cos⁡(x))log⁡(x)cos⁡2(x)−(3xsin⁡(x)−cos⁡(x))log⁡(x)xcos⁡(x)+2(3xsin⁡(x)−cos⁡(x))xcos⁡(x)+log⁡(xcos⁡3(x))x)x+6(−2xsin⁡2(x)+xcos⁡2(x)+2sin⁡(x)cos⁡(x))log⁡(x)xcos⁡2(x)−9(−2xsin⁡2(x)+xcos⁡2(x)+2sin⁡(x)cos⁡(x))xcos⁡2(x)−2(3xsin⁡(x)−cos⁡(x))log⁡(x)x2cos⁡(x)+6(3xsin⁡(x)−cos⁡(x))x2cos⁡(x)−((3xsin⁡(x)−cos⁡(x))log⁡(x)cos⁡(x)−log⁡(xcos⁡3(x)))3x2+2log⁡(xcos⁡3(x))x2)x\frac{\left(x \cos^{3}{\left(x \right)}\right)^{\log{\left(x \right)}} \left(- \frac{12 \left(3 x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \log{\left(x \right)} \sin^{2}{\left(x \right)}}{\cos^{3}{\left(x \right)}} - \frac{3 \left(3 x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \log{\left(x \right)}}{\cos{\left(x \right)}} - \frac{18 \left(- 2 x \sin^{2}{\left(x \right)} + x \cos^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)}\right) \log{\left(x \right)} \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}} - \frac{3 \left(2 x \sin^{3}{\left(x \right)} - 7 x \sin{\left(x \right)} \cos^{2}{\left(x \right)} - 6 \sin^{2}{\left(x \right)} \cos{\left(x \right)} + 3 \cos^{3}{\left(x \right)}\right) \log{\left(x \right)}}{\cos^{3}{\left(x \right)}} + \frac{6 \left(3 x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \log{\left(x \right)} \sin{\left(x \right)}}{x \cos^{2}{\left(x \right)}} - \frac{9 \left(3 x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \sin{\left(x \right)}}{x \cos^{2}{\left(x \right)}} + \frac{3 \left(\frac{\left(3 x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \log{\left(x \right)}}{\cos{\left(x \right)}} - \log{\left(x \cos^{3}{\left(x \right)} \right)}\right) \left(\frac{3 \left(3 x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \log{\left(x \right)} \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{3 \left(- 2 x \sin^{2}{\left(x \right)} + x \cos^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)}\right) \log{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{\left(3 x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \log{\left(x \right)}}{x \cos{\left(x \right)}} + \frac{2 \left(3 x \sin{\left(x \right)} - \cos{\left(x \right)}\right)}{x \cos{\left(x \right)}} + \frac{\log{\left(x \cos^{3}{\left(x \right)} \right)}}{x}\right)}{x} + \frac{6 \left(- 2 x \sin^{2}{\left(x \right)} + x \cos^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)}\right) \log{\left(x \right)}}{x \cos^{2}{\left(x \right)}} - \frac{9 \left(- 2 x \sin^{2}{\left(x \right)} + x \cos^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)}\right)}{x \cos^{2}{\left(x \right)}} - \frac{2 \left(3 x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \log{\left(x \right)}}{x^{2} \cos{\left(x \right)}} + \frac{6 \left(3 x \sin{\left(x \right)} - \cos{\left(x \right)}\right)}{x^{2} \cos{\left(x \right)}} - \frac{\left(\frac{\left(3 x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \log{\left(x \right)}}{\cos{\left(x \right)}} - \log{\left(x \cos^{3}{\left(x \right)} \right)}\right)^{3}}{x^{2}} + \frac{2 \log{\left(x \cos^{3}{\left(x \right)} \right)}}{x^{2}}\right)}{x}
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Gráfico
Derivada de y=((cos^3)x)^(lnx)
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