Derivada de x^sinx^x+3^2^x

Ha introducido [src]

    x       / x\
 sin (x)    \2 /
x        + 3

3^{2^{x}} + x^{\sin^{x}{\left(x \right)}}

x^(sin(x)^x) + 3^(2^x)

Solución detallada

diferenciamos $3^{2^{x}} + x^{\sin^{x}{\left(x \right)}}$ miembro por miembro:
1. No logro encontrar los pasos en la búsqueda de esta derivada.
  
  Perola derivada
  
  $\left(\log{\left(\sin^{x}{\left(x \right)} \right)} + 1\right) \left(\sin^{x}{\left(x \right)}\right)^{\sin^{x}{\left(x \right)}}$
2. Sustituimos $u = 2^{x}$ .
3. $\frac{d}{d u} 3^{u} = 3^{u} \log{\left(3 \right)}$
4. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} 2^{x}$ :
  1. $\frac{d}{d x} 2^{x} = 2^{x} \log{\left(2 \right)}$
  Como resultado de la secuencia de reglas:
  
  $2^{x} 3^{2^{x}} \log{\left(2 \right)} \log{\left(3 \right)}$
Como resultado de: $2^{x} 3^{2^{x}} \log{\left(2 \right)} \log{\left(3 \right)} + \left(\log{\left(\sin^{x}{\left(x \right)} \right)} + 1\right) \left(\sin^{x}{\left(x \right)}\right)^{\sin^{x}{\left(x \right)}}$

Respuesta:

$2^{x} 3^{2^{x}} \log{\left(2 \right)} \log{\left(3 \right)} + \left(\log{\left(\sin^{x}{\left(x \right)} \right)} + 1\right) \left(\sin^{x}{\left(x \right)}\right)^{\sin^{x}{\left(x \right)}}$

Primera derivada [src]

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

2^{x} 3^{2^{x}} \log{\left(2 \right)} \log{\left(3 \right)} + 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                 x            |                               2          /                    2   \          2*|-------- + log(sin(x))||       / x\                        / x\                
 sin (x) /1   /x*cos(x)              \       \     2*x       sin (x)    x    |  1    /x*cos(x)              \           |    2*cos(x)   x*cos (x)|            \ sin(x)               /|    x  \2 /    2              2*x  \2 /    2       2   
x       *|- + |-------- + log(sin(x))|*log(x)| *sin   (x) + x       *sin (x)*|- -- + |-------- + log(sin(x))| *log(x) - |x - -------- + ---------|*log(x) + --------------------------| + 2 *3    *log (2)*log(3) + 2   *3    *log (2)*log (3)
         \x   \ sin(x)               /       /                               |   2   \ sin(x)               /           |     sin(x)        2    |                      x             |                                                       
                                                                             \  x                                       \                sin (x) /                                    /

2^{2 x} 3^{2^{x}} \log{\left(2 \right)}^{2} \log{\left(3 \right)}^{2} + 2^{x} 3^{2^{x}} \log{\left(2 \right)}^{2} \log{\left(3 \right)} + 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)} + \frac{1}{x}\right)^{2} \sin^{2 x}{\left(x \right)} + 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)^{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                 x            |                             3          /          2             3                \            |     sin(x)        2    |   3*|-------- + log(sin(x))|   3*|-------- + log(sin(x))|                               /                    2   \       |       / x\                        / x\                           / x\                         x                                                    |                               2          /                    2   \          2*|-------- + log(sin(x))||
 sin (x) /1   /x*cos(x)              \       \     3*x       sin (x)    x    |2    /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  \2 /    3              3*x  \2 /    3       3         2*x  \2 /    3       2         sin (x)    2*x    /1   /x*cos(x)              \       \ |  1    /x*cos(x)              \           |    2*cos(x)   x*cos (x)|            \ sin(x)               /|
x       *|- + |-------- + log(sin(x))|*log(x)| *sin   (x) + x       *sin (x)*|-- + |-------- + log(sin(x))| *log(x) + |-3 - --------- + ----------- + ----------|*log(x) - ---------------------------- - -------------------------- + --------------------------- - 3*|-------- + log(sin(x))|*|x - -------- + ---------|*log(x)| + 2 *3    *log (2)*log(3) + 2   *3    *log (2)*log (3) + 3*2   *3    *log (2)*log (3) + 3*x       *sin   (x)*|- + |-------- + log(sin(x))|*log(x)|*|- -- + |-------- + log(sin(x))| *log(x) - |x - -------- + ---------|*log(x) + --------------------------|
         \x   \ sin(x)               /       /                               | 3   \ 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) /                                    /

2^{3 x} 3^{2^{x}} \log{\left(2 \right)}^{3} \log{\left(3 \right)}^{3} + 3 \cdot 2^{2 x} 3^{2^{x}} \log{\left(2 \right)}^{3} \log{\left(3 \right)}^{2} + 2^{x} 3^{2^{x}} \log{\left(2 \right)}^{3} \log{\left(3 \right)} + 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)} + \frac{1}{x}\right)^{3} \sin^{3 x}{\left(x \right)} + 3 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)} + \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^{2 x}{\left(x \right)} + 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)^{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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Derivada de x^sinx^x+3^2^x

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