Derivada de y=((x)^sinx)(sin)^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

Se aplica la regla de la derivada de una multiplicación:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = x^{\sin{\left(x \right)}}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
1. No logro encontrar los pasos en la búsqueda de esta derivada.
  
  Perola derivada
  
  $\left(\log{\left(\sin{\left(x \right)} \right)} + 1\right) \sin^{\sin{\left(x \right)}}{\left(x \right)}$
$g{\left(x \right)} = \sin^{x}{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
1. 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: $x^{x} x^{\sin{\left(x \right)}} \left(\log{\left(x \right)} + 1\right) + \left(\log{\left(\sin{\left(x \right)} \right)} + 1\right) \sin^{x}{\left(x \right)} \sin^{\sin{\left(x \right)}}{\left(x \right)}$
Simplificamos:

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

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

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

Simplificar

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Derivada de y=((x)^sinx)(sin)^x

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