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y=e^sqrt(x/lnx)
Derivada de la función/ x/lnx/ y=e^sqrt(x/lnx)

Derivada de y=e^sqrt(x/lnx)

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

Ha introducido [src] 
     ________
    /   x    
   /  ------ 
 \/   log(x) 
E
exlog⁡(x)e^{\sqrt{\frac{x}{\log{\left(x \right)}}}} 
E^(sqrt(x/log(x)))
Solución detallada

  1. Sustituimos u=xlog⁡(x)u = \sqrt{\frac{x}{\log{\left(x \right)}}}.

  2. Derivado eue^{u} es.

  3. Luego se aplica una cadena de reglas. Multiplicamos por ddxxlog⁡(x)\frac{d}{d x} \sqrt{\frac{x}{\log{\left(x \right)}}}:

    1. Sustituimos u=xlog⁡(x)u = \frac{x}{\log{\left(x \right)}}.

    2. Según el principio, aplicamos: u\sqrt{u} tenemos 12u\frac{1}{2 \sqrt{u}}

    3. Luego se aplica una cadena de reglas. Multiplicamos por ddxxlog⁡(x)\frac{d}{d x} \frac{x}{\log{\left(x \right)}}:

      1. Se aplica la regla de la derivada parcial:

        ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}

        f(x)=xf{\left(x \right)} = x y g(x)=log⁡(x)g{\left(x \right)} = \log{\left(x \right)}.

        Para calcular ddxf(x)\frac{d}{d x} f{\left(x \right)}:

        1. Según el principio, aplicamos: xx tenemos 11

        Para calcular ddxg(x)\frac{d}{d x} g{\left(x \right)}:

        1. Derivado log⁡(x)\log{\left(x \right)} es 1x\frac{1}{x}.

        Ahora aplicamos la regla de la derivada de una divesión:

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

      Como resultado de la secuencia de reglas:

      log⁡(x)−12xlog⁡(x)log⁡(x)2\frac{\log{\left(x \right)} - 1}{2 \sqrt{\frac{x}{\log{\left(x \right)}}} \log{\left(x \right)}^{2}}

    Como resultado de la secuencia de reglas:

    (log⁡(x)−1)exlog⁡(x)2xlog⁡(x)log⁡(x)2\frac{\left(\log{\left(x \right)} - 1\right) e^{\sqrt{\frac{x}{\log{\left(x \right)}}}}}{2 \sqrt{\frac{x}{\log{\left(x \right)}}} \log{\left(x \right)}^{2}}

Respuesta:

(log⁡(x)−1)exlog⁡(x)2xlog⁡(x)log⁡(x)2\frac{\left(\log{\left(x \right)} - 1\right) e^{\sqrt{\frac{x}{\log{\left(x \right)}}}}}{2 \sqrt{\frac{x}{\log{\left(x \right)}}} \log{\left(x \right)}^{2}}

Gráfica
02468-8-6-4-2-1010-500500
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Primera derivada [src] 
                                         ________       
                                        /   x           
    ________                           /  ------        
   /   x     /   1           1    \  \/   log(x)        
  /  ------ *|-------- - ---------|*e            *log(x)
\/   log(x)  |2*log(x)        2   |                     
             \           2*log (x)/                     
--------------------------------------------------------
                           x
xlog⁡(x)(12log⁡(x)−12log⁡(x)2)exlog⁡(x)log⁡(x)x\frac{\sqrt{\frac{x}{\log{\left(x \right)}}} \left(\frac{1}{2 \log{\left(x \right)}} - \frac{1}{2 \log{\left(x \right)}^{2}}\right) e^{\sqrt{\frac{x}{\log{\left(x \right)}}}} \log{\left(x \right)}}{x}
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Segunda derivada [src] 
/            2       ________             2         ________                      ________                      ________             \      ________
|/      1   \       /   x     /      1   \         /   x     /      1   \        /   x     /      2   \        /   x     /      1   \|     /   x    
||1 - ------|      /  ------ *|1 - ------|    2*  /  ------ *|1 - ------|   2*  /  ------ *|1 - ------|   2*  /  ------ *|1 - ------||    /  ------ 
|\    log(x)/    \/   log(x)  \    log(x)/      \/   log(x)  \    log(x)/     \/   log(x)  \    log(x)/     \/   log(x)  \    log(x)/|  \/   log(x) 
|------------- + -------------------------- - --------------------------- - --------------------------- + ---------------------------|*e            
\    log(x)                  x                             x                          x*log(x)                      x*log(x)         /              
----------------------------------------------------------------------------------------------------------------------------------------------------
                                                                        4*x
((1−1log⁡(x))2log⁡(x)−2xlog⁡(x)(1−2log⁡(x))xlog⁡(x)+xlog⁡(x)(1−1log⁡(x))2x−2xlog⁡(x)(1−1log⁡(x))x+2xlog⁡(x)(1−1log⁡(x))xlog⁡(x))exlog⁡(x)4x\frac{\left(\frac{\left(1 - \frac{1}{\log{\left(x \right)}}\right)^{2}}{\log{\left(x \right)}} - \frac{2 \sqrt{\frac{x}{\log{\left(x \right)}}} \left(1 - \frac{2}{\log{\left(x \right)}}\right)}{x \log{\left(x \right)}} + \frac{\sqrt{\frac{x}{\log{\left(x \right)}}} \left(1 - \frac{1}{\log{\left(x \right)}}\right)^{2}}{x} - \frac{2 \sqrt{\frac{x}{\log{\left(x \right)}}} \left(1 - \frac{1}{\log{\left(x \right)}}\right)}{x} + \frac{2 \sqrt{\frac{x}{\log{\left(x \right)}}} \left(1 - \frac{1}{\log{\left(x \right)}}\right)}{x \log{\left(x \right)}}\right) e^{\sqrt{\frac{x}{\log{\left(x \right)}}}}}{4 x}
Simplificar
Tercera derivada [src] 
/                                                                                                                                                                                                                                   ________                                                                                                                                                  \              
|            2               2               3       ________                      ________             2                                     ________             3       ________             3       ________                   /   x     /       6   \       ________                      ________                      ________             2         ________                          |      ________
|/      1   \    /      1   \    /      1   \       /   x     /      1   \        /   x     /      1   \      /      1   \ /      2   \      /   x     /      1   \       /   x     /      1   \       /   x     /      2   \     /  ------ *|1 - -------|      /   x     /      2   \        /   x     /      1   \        /   x     /      1   \         /   x     /      1   \ /      2   \|     /   x    
||1 - ------|    |1 - ------|    |1 - ------|      /  ------ *|1 - ------|   3*  /  ------ *|1 - ------|    3*|1 - ------|*|1 - ------|     /  ------ *|1 - ------|      /  ------ *|1 - ------|      /  ------ *|1 - ------|   \/   log(x)  |       2   |     /  ------ *|1 - ------|   3*  /  ------ *|1 - ------|   3*  /  ------ *|1 - ------|    3*  /  ------ *|1 - ------|*|1 - ------||    /  ------ 
|\    log(x)/    \    log(x)/    \    log(x)/    \/   log(x)  \    log(x)/     \/   log(x)  \    log(x)/      \    log(x)/ \    log(x)/   \/   log(x)  \    log(x)/    \/   log(x)  \    log(x)/    \/   log(x)  \    log(x)/                \    log (x)/   \/   log(x)  \    log(x)/     \/   log(x)  \    log(x)/     \/   log(x)  \    log(x)/      \/   log(x)  \    log(x)/ \    log(x)/|  \/   log(x) 
|------------- - ------------- + ------------- + ------------------------- - ---------------------------- - --------------------------- + -------------------------- + -------------------------- + ------------------------- + -------------------------- - ------------------------- - --------------------------- + ---------------------------- - ----------------------------------------|*e            
|       2           2*log(x)        8*log(x)                 x                           4*x                              2                          8*x                        8*log(x)                     x*log(x)                   2*x*log(x)                        2                       2*x*log(x)                    4*x*log(x)                           4*x*log(x)               |              
\  2*log (x)                                                                                                         4*log (x)                                                                                                                                       x*log (x)                                                                                                                /              
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                       2                                                                                                                                                                                                     
                                                                                                                                                                                                      x
(xlog⁡(x)(1−1log⁡(x))38log⁡(x)−3(1−2log⁡(x))(1−1log⁡(x))4log⁡(x)2+(1−1log⁡(x))38log⁡(x)−(1−1log⁡(x))22log⁡(x)+(1−1log⁡(x))22log⁡(x)2+xlog⁡(x)(1−6log⁡(x)2)2xlog⁡(x)−3xlog⁡(x)(1−2log⁡(x))(1−1log⁡(x))4xlog⁡(x)+xlog⁡(x)(1−2log⁡(x))xlog⁡(x)−xlog⁡(x)(1−2log⁡(x))xlog⁡(x)2+xlog⁡(x)(1−1log⁡(x))38x−3xlog⁡(x)(1−1log⁡(x))24x+3xlog⁡(x)(1−1log⁡(x))24xlog⁡(x)+xlog⁡(x)(1−1log⁡(x))x−3xlog⁡(x)(1−1log⁡(x))2xlog⁡(x))exlog⁡(x)x2\frac{\left(\frac{\sqrt{\frac{x}{\log{\left(x \right)}}} \left(1 - \frac{1}{\log{\left(x \right)}}\right)^{3}}{8 \log{\left(x \right)}} - \frac{3 \left(1 - \frac{2}{\log{\left(x \right)}}\right) \left(1 - \frac{1}{\log{\left(x \right)}}\right)}{4 \log{\left(x \right)}^{2}} + \frac{\left(1 - \frac{1}{\log{\left(x \right)}}\right)^{3}}{8 \log{\left(x \right)}} - \frac{\left(1 - \frac{1}{\log{\left(x \right)}}\right)^{2}}{2 \log{\left(x \right)}} + \frac{\left(1 - \frac{1}{\log{\left(x \right)}}\right)^{2}}{2 \log{\left(x \right)}^{2}} + \frac{\sqrt{\frac{x}{\log{\left(x \right)}}} \left(1 - \frac{6}{\log{\left(x \right)}^{2}}\right)}{2 x \log{\left(x \right)}} - \frac{3 \sqrt{\frac{x}{\log{\left(x \right)}}} \left(1 - \frac{2}{\log{\left(x \right)}}\right) \left(1 - \frac{1}{\log{\left(x \right)}}\right)}{4 x \log{\left(x \right)}} + \frac{\sqrt{\frac{x}{\log{\left(x \right)}}} \left(1 - \frac{2}{\log{\left(x \right)}}\right)}{x \log{\left(x \right)}} - \frac{\sqrt{\frac{x}{\log{\left(x \right)}}} \left(1 - \frac{2}{\log{\left(x \right)}}\right)}{x \log{\left(x \right)}^{2}} + \frac{\sqrt{\frac{x}{\log{\left(x \right)}}} \left(1 - \frac{1}{\log{\left(x \right)}}\right)^{3}}{8 x} - \frac{3 \sqrt{\frac{x}{\log{\left(x \right)}}} \left(1 - \frac{1}{\log{\left(x \right)}}\right)^{2}}{4 x} + \frac{3 \sqrt{\frac{x}{\log{\left(x \right)}}} \left(1 - \frac{1}{\log{\left(x \right)}}\right)^{2}}{4 x \log{\left(x \right)}} + \frac{\sqrt{\frac{x}{\log{\left(x \right)}}} \left(1 - \frac{1}{\log{\left(x \right)}}\right)}{x} - \frac{3 \sqrt{\frac{x}{\log{\left(x \right)}}} \left(1 - \frac{1}{\log{\left(x \right)}}\right)}{2 x \log{\left(x \right)}}\right) e^{\sqrt{\frac{x}{\log{\left(x \right)}}}}}{x^{2}}
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Gráfico
Derivada de y=e^sqrt(x/lnx)
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