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

Derivada de x^sinx^x

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

Gráfico:

interior superior

Definida a trozos:

Solución

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

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

    Perola derivada

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

Respuesta:

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

Primera derivada [src] 
    x    /   x                                             \
 sin (x) |sin (x)      x    /x*cos(x)              \       |
x       *|------- + sin (x)*|-------- + log(sin(x))|*log(x)|
         \   x              \ sin(x)               /       /
xsin⁡x(x)((xcos⁡(x)sin⁡(x)+log⁡(sin⁡(x)))log⁡(x)sin⁡x(x)+sin⁡x(x)x)x^{\sin^{x}{\left(x \right)}} \left(\left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \log{\left(x \right)} \sin^{x}{\left(x \right)} + \frac{\sin^{x}{\left(x \right)}}{x}\right)
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Segunda derivada [src] 
                 /                                                                                                                                 /x*cos(x)              \\
    x            |                                            2                                   2          /                    2   \          2*|-------- + log(sin(x))||
 sin (x)    x    |  1    /1   /x*cos(x)              \       \     x      /x*cos(x)              \           |    2*cos(x)   x*cos (x)|            \ sin(x)               /|
x       *sin (x)*|- -- + |- + |-------- + log(sin(x))|*log(x)| *sin (x) + |-------- + log(sin(x))| *log(x) - |x - -------- + ---------|*log(x) + --------------------------|
                 |   2   \x   \ sin(x)               /       /            \ sin(x)               /           |     sin(x)        2    |                      x             |
                 \  x                                                                                        \                sin (x) /                                    /
xsin⁡x(x)(((xcos⁡(x)sin⁡(x)+log⁡(sin⁡(x)))log⁡(x)+1x)2sin⁡x(x)+(xcos⁡(x)sin⁡(x)+log⁡(sin⁡(x)))2log⁡(x)−(x+xcos⁡2(x)sin⁡2(x)−2cos⁡(x)sin⁡(x))log⁡(x)+2(xcos⁡(x)sin⁡(x)+log⁡(sin⁡(x)))x−1x2)sin⁡x(x)x^{\sin^{x}{\left(x \right)}} \left(\left(\left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \log{\left(x \right)} + \frac{1}{x}\right)^{2} \sin^{x}{\left(x \right)} + \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right)^{2} \log{\left(x \right)} - \left(x + \frac{x \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) \log{\left(x \right)} + \frac{2 \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right)}{x} - \frac{1}{x^{2}}\right) \sin^{x}{\left(x \right)}
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Tercera derivada [src] 
                 /                                                                                                                                                  /                    2   \                                                                                                                                                                                                                                                                                       \
                 |                                                                                                                                                  |    2*cos(x)   x*cos (x)|                                                          2                                                                                                                                                                                                                            |
                 |                                                                                                                                                3*|x - -------- + ---------|     /x*cos(x)              \     /x*cos(x)              \                                                                                                                   /                                                                                /x*cos(x)              \\|
    x            |                                          3                                     3          /          2             3                \            |     sin(x)        2    |   3*|-------- + log(sin(x))|   3*|-------- + log(sin(x))|                               /                    2   \                                                          |                               2          /                    2   \          2*|-------- + log(sin(x))|||
 sin (x)    x    |2    /1   /x*cos(x)              \       \     2*x      /x*cos(x)              \           |     3*cos (x)   2*x*cos (x)   2*x*cos(x)|            \                sin (x) /     \ sin(x)               /     \ sin(x)               /      /x*cos(x)              \ |    2*cos(x)   x*cos (x)|               x    /1   /x*cos(x)              \       \ |  1    /x*cos(x)              \           |    2*cos(x)   x*cos (x)|            \ sin(x)               /||
x       *sin (x)*|-- + |- + |-------- + log(sin(x))|*log(x)| *sin   (x) + |-------- + log(sin(x))| *log(x) + |-3 - --------- + ----------- + ----------|*log(x) - ---------------------------- - -------------------------- + --------------------------- - 3*|-------- + log(sin(x))|*|x - -------- + ---------|*log(x) + 3*sin (x)*|- + |-------- + log(sin(x))|*log(x)|*|- -- + |-------- + log(sin(x))| *log(x) - |x - -------- + ---------|*log(x) + --------------------------||
                 | 3   \x   \ sin(x)               /       /              \ sin(x)               /           |         2            3          sin(x)  |                       x                              2                            x                  \ sin(x)               / |     sin(x)        2    |                    \x   \ sin(x)               /       / |   2   \ sin(x)               /           |     sin(x)        2    |                      x             ||
                 \x                                                                                          \      sin (x)      sin (x)               /                                                     x                                                                         \                sin (x) /                                                          \  x                                       \                sin (x) /                                    //
xsin⁡x(x)(((xcos⁡(x)sin⁡(x)+log⁡(sin⁡(x)))log⁡(x)+1x)3sin⁡2x(x)+3((xcos⁡(x)sin⁡(x)+log⁡(sin⁡(x)))log⁡(x)+1x)((xcos⁡(x)sin⁡(x)+log⁡(sin⁡(x)))2log⁡(x)−(x+xcos⁡2(x)sin⁡2(x)−2cos⁡(x)sin⁡(x))log⁡(x)+2(xcos⁡(x)sin⁡(x)+log⁡(sin⁡(x)))x−1x2)sin⁡x(x)+(xcos⁡(x)sin⁡(x)+log⁡(sin⁡(x)))3log⁡(x)−3(xcos⁡(x)sin⁡(x)+log⁡(sin⁡(x)))(x+xcos⁡2(x)sin⁡2(x)−2cos⁡(x)sin⁡(x))log⁡(x)+(2xcos⁡(x)sin⁡(x)+2xcos⁡3(x)sin⁡3(x)−3−3cos⁡2(x)sin⁡2(x))log⁡(x)+3(xcos⁡(x)sin⁡(x)+log⁡(sin⁡(x)))2x−3(x+xcos⁡2(x)sin⁡2(x)−2cos⁡(x)sin⁡(x))x−3(xcos⁡(x)sin⁡(x)+log⁡(sin⁡(x)))x2+2x3)sin⁡x(x)x^{\sin^{x}{\left(x \right)}} \left(\left(\left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \log{\left(x \right)} + \frac{1}{x}\right)^{3} \sin^{2 x}{\left(x \right)} + 3 \left(\left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \log{\left(x \right)} + \frac{1}{x}\right) \left(\left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right)^{2} \log{\left(x \right)} - \left(x + \frac{x \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) \log{\left(x \right)} + \frac{2 \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right)}{x} - \frac{1}{x^{2}}\right) \sin^{x}{\left(x \right)} + \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right)^{3} \log{\left(x \right)} - 3 \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \left(x + \frac{x \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) \log{\left(x \right)} + \left(\frac{2 x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{2 x \cos^{3}{\left(x \right)}}{\sin^{3}{\left(x \right)}} - 3 - \frac{3 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \log{\left(x \right)} + \frac{3 \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right)^{2}}{x} - \frac{3 \left(x + \frac{x \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right)}{x} - \frac{3 \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right)}{x^{2}} + \frac{2}{x^{3}}\right) \sin^{x}{\left(x \right)}
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