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Derivada de la función/ e^x^2/ y=e^x/ y=e^x^2^-x

Derivada de y=e^x^2^-x

Función f() - derivada -er orden en el punto
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Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 / / -x\\
 | \2  /|
 \x     /
E
ex2−xe^{x^{2^{- x}}} 
E^(x^(2^(-x)))
Solución detallada

  1. Sustituimos u=x2−xu = x^{2^{- x}}.

  2. Derivado eue^{u} es.

  3. Luego se aplica una cadena de reglas. Multiplicamos por ddxx2−x\frac{d}{d x} x^{2^{- x}}:

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

      Perola derivada

      2−2−xx(log⁡(2−x)+1)2^{- 2^{- x} x} \left(\log{\left(2^{- x} \right)} + 1\right)

    Como resultado de la secuencia de reglas:

    2−2−xx(log⁡(2−x)+1)ex2−x2^{- 2^{- x} x} \left(\log{\left(2^{- x} \right)} + 1\right) e^{x^{2^{- x}}}

  4. Simplificamos:

    2−2−xx(−xlog⁡(2)+1)ex2−x2^{- 2^{- x} x} \left(- x \log{\left(2 \right)} + 1\right) e^{x^{2^{- x}}}

Respuesta:

2−2−xx(−xlog⁡(2)+1)ex2−x2^{- 2^{- x} x} \left(- x \log{\left(2 \right)} + 1\right) e^{x^{2^{- x}}}

Primera derivada [src] 
                                  / / -x\\
 / -x\ / -x                    \  | \2  /|
 \2  / |2      -x              |  \x     /
x     *|--- - 2  *log(2)*log(x)|*e        
       \ x                     /
x2−x(−2−xlog⁡(2)log⁡(x)+2−xx)ex2−xx^{2^{- x}} \left(- 2^{- x} \log{\left(2 \right)} \log{\left(x \right)} + \frac{2^{- x}}{x}\right) e^{x^{2^{- x}}}
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Segunda derivada [src] 
                                                                                                                / / -x\\
     / -x\ /                                2                                    / -x\                      2\  | \2  /|
 -x  \2  / |  1     -x /  1                \       2             2*log(2)    -x  \2  / /  1                \ |  \x     /
2  *x     *|- -- + 2  *|- - + log(2)*log(x)|  + log (2)*log(x) - -------- + 2  *x     *|- - + log(2)*log(x)| |*e        
           |   2       \  x                /                        x                  \  x                / |          
           \  x                                                                                              /
2−xx2−x(log⁡(2)2log⁡(x)−2log⁡(2)x−1x2+2−xx2−x(log⁡(2)log⁡(x)−1x)2+2−x(log⁡(2)log⁡(x)−1x)2)ex2−x2^{- x} x^{2^{- x}} \left(\log{\left(2 \right)}^{2} \log{\left(x \right)} - \frac{2 \log{\left(2 \right)}}{x} - \frac{1}{x^{2}} + 2^{- x} x^{2^{- x}} \left(\log{\left(2 \right)} \log{\left(x \right)} - \frac{1}{x}\right)^{2} + 2^{- x} \left(\log{\left(2 \right)} \log{\left(x \right)} - \frac{1}{x}\right)^{2}\right) e^{x^{2^{- x}}}
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Tercera derivada [src] 
                                                                                                                                                                                                                                                                                                           / / -x\\
     / -x\ /                                3                         2                           -x                      3            / -x\                      3                                                                         / -x\                                                       \  | \2  /|
 -x  \2  / |2     -2*x /  1                \       3             3*log (2)   3*log(2)    -2*x  2*2   /  1                \       -2*x  \2  / /  1                \       -x /  1                \ /1       2             2*log(2)\      -x  \2  / /  1                \ /1       2             2*log(2)\|  \x     /
2  *x     *|-- - 2    *|- - + log(2)*log(x)|  - log (2)*log(x) + --------- + -------- - 2    *x     *|- - + log(2)*log(x)|  - 3*2    *x     *|- - + log(2)*log(x)|  + 3*2  *|- - + log(2)*log(x)|*|-- - log (2)*log(x) + --------| + 3*2  *x     *|- - + log(2)*log(x)|*|-- - log (2)*log(x) + --------||*e        
           | 3         \  x                /                         x           2                   \  x                /                   \  x                /          \  x                / | 2                       x    |                \  x                / | 2                       x    ||          
           \x                                                                   x                                                                                                                 \x                             /                                      \x                             //
2−xx2−x(−log⁡(2)3log⁡(x)+3log⁡(2)2x+3log⁡(2)x2+2x3+3⋅2−xx2−x(log⁡(2)log⁡(x)−1x)(−log⁡(2)2log⁡(x)+2log⁡(2)x+1x2)+3⋅2−x(log⁡(2)log⁡(x)−1x)(−log⁡(2)2log⁡(x)+2log⁡(2)x+1x2)−2−2xx2⋅2−x(log⁡(2)log⁡(x)−1x)3−3⋅2−2xx2−x(log⁡(2)log⁡(x)−1x)3−2−2x(log⁡(2)log⁡(x)−1x)3)ex2−x2^{- x} x^{2^{- x}} \left(- \log{\left(2 \right)}^{3} \log{\left(x \right)} + \frac{3 \log{\left(2 \right)}^{2}}{x} + \frac{3 \log{\left(2 \right)}}{x^{2}} + \frac{2}{x^{3}} + 3 \cdot 2^{- x} x^{2^{- x}} \left(\log{\left(2 \right)} \log{\left(x \right)} - \frac{1}{x}\right) \left(- \log{\left(2 \right)}^{2} \log{\left(x \right)} + \frac{2 \log{\left(2 \right)}}{x} + \frac{1}{x^{2}}\right) + 3 \cdot 2^{- x} \left(\log{\left(2 \right)} \log{\left(x \right)} - \frac{1}{x}\right) \left(- \log{\left(2 \right)}^{2} \log{\left(x \right)} + \frac{2 \log{\left(2 \right)}}{x} + \frac{1}{x^{2}}\right) - 2^{- 2 x} x^{2 \cdot 2^{- x}} \left(\log{\left(2 \right)} \log{\left(x \right)} - \frac{1}{x}\right)^{3} - 3 \cdot 2^{- 2 x} x^{2^{- x}} \left(\log{\left(2 \right)} \log{\left(x \right)} - \frac{1}{x}\right)^{3} - 2^{- 2 x} \left(\log{\left(2 \right)} \log{\left(x \right)} - \frac{1}{x}\right)^{3}\right) e^{x^{2^{- x}}}
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