Derivada de y=(sinx)^lnx

Ha introducido [src]

   log(x)   
sin      (x)

$$\sin^{\log{\left(x \right)}}{\left(x \right)}$$

sin(x)^log(x)

Solución detallada

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Perola derivada

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

   log(x)    /log(sin(x))   cos(x)*log(x)\
sin      (x)*|----------- + -------------|
             \     x            sin(x)   /

$$\left(\frac{\log{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{\log{\left(\sin{\left(x \right)} \right)}}{x}\right) \sin^{\log{\left(x \right)}}{\left(x \right)}$$

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Segunda derivada [src]

             /                             2                             2                     \
   log(x)    |/log(sin(x))   cos(x)*log(x)\             log(sin(x))   cos (x)*log(x)   2*cos(x)|
sin      (x)*||----------- + -------------|  - log(x) - ----------- - -------------- + --------|
             |\     x            sin(x)   /                   2             2          x*sin(x)|
             \                                               x           sin (x)               /

$$\left(\left(\frac{\log{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{\log{\left(\sin{\left(x \right)} \right)}}{x}\right)^{2} - \log{\left(x \right)} - \frac{\log{\left(x \right)} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{2 \cos{\left(x \right)}}{x \sin{\left(x \right)}} - \frac{\log{\left(\sin{\left(x \right)} \right)}}{x^{2}}\right) \sin^{\log{\left(x \right)}}{\left(x \right)}$$

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Tercera derivada [src]

             /                             3                                       /                 2                              \                        2                       3                            \
   log(x)    |/log(sin(x))   cos(x)*log(x)\    3     /log(sin(x))   cos(x)*log(x)\ |log(sin(x))   cos (x)*log(x)   2*cos(x)         |   2*log(sin(x))   3*cos (x)    3*cos(x)   2*cos (x)*log(x)   2*cos(x)*log(x)|
sin      (x)*||----------- + -------------|  - - - 3*|----------- + -------------|*|----------- + -------------- - -------- + log(x)| + ------------- - --------- - --------- + ---------------- + ---------------|
             |\     x            sin(x)   /    x     \     x            sin(x)   / |      2             2          x*sin(x)         |          3             2       2                 3                sin(x)    |
             \                                                                     \     x           sin (x)                        /         x         x*sin (x)   x *sin(x)       sin (x)                       /

$$\left(\left(\frac{\log{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{\log{\left(\sin{\left(x \right)} \right)}}{x}\right)^{3} - 3 \left(\frac{\log{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{\log{\left(\sin{\left(x \right)} \right)}}{x}\right) \left(\log{\left(x \right)} + \frac{\log{\left(x \right)} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{2 \cos{\left(x \right)}}{x \sin{\left(x \right)}} + \frac{\log{\left(\sin{\left(x \right)} \right)}}{x^{2}}\right) + \frac{2 \log{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2 \log{\left(x \right)} \cos^{3}{\left(x \right)}}{\sin^{3}{\left(x \right)}} - \frac{3}{x} - \frac{3 \cos^{2}{\left(x \right)}}{x \sin^{2}{\left(x \right)}} - \frac{3 \cos{\left(x \right)}}{x^{2} \sin{\left(x \right)}} + \frac{2 \log{\left(\sin{\left(x \right)} \right)}}{x^{3}}\right) \sin^{\log{\left(x \right)}}{\left(x \right)}$$

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Gráfico

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Derivada de y=(sinx)^lnx

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Solución