Derivada de (x)^sinx+(sinx)^x

Solución

Ha introducido [src]

 sin(x)      x   
x       + sin (x)

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

x^sin(x) + sin(x)^x

Solución detallada

diferenciamos $x^{\sin{\left(x \right)}} + \sin^{x}{\left(x \right)}$ miembro por miembro:
1. No logro encontrar los pasos en la búsqueda de esta derivada.
  
  Perola derivada
  
  $\left(\log{\left(\sin{\left(x \right)} \right)} + 1\right) \sin^{\sin{\left(x \right)}}{\left(x \right)}$
2. No logro encontrar los pasos en la búsqueda de esta derivada.
  
  Perola derivada
  
  $x^{x} \left(\log{\left(x \right)} + 1\right)$
Como resultado de: $x^{x} \left(\log{\left(x \right)} + 1\right) + \left(\log{\left(\sin{\left(x \right)} \right)} + 1\right) \sin^{\sin{\left(x \right)}}{\left(x \right)}$

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

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Tercera derivada [src]

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

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

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Derivada de (x)^sinx+(sinx)^x

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