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Derivada de z=e^((t^2/ln(t))^1/2)

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

Gráfico:

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Definida a trozos:

Solución

Ha introducido [src]
      ________
     /    2   
    /    t    
   /   ------ 
 \/    log(t) 
E             
$$e^{\sqrt{\frac{t^{2}}{\log{\left(t \right)}}}}$$
E^(sqrt(t^2/log(t)))
Solución detallada
  1. Sustituimos .

  2. Derivado es.

  3. Luego se aplica una cadena de reglas. Multiplicamos por :

    1. Sustituimos .

    2. Según el principio, aplicamos: tenemos

    3. Luego se aplica una cadena de reglas. Multiplicamos por :

      1. Se aplica la regla de la derivada parcial:

        y .

        Para calcular :

        1. Según el principio, aplicamos: tenemos

        Para calcular :

        1. Derivado es .

        Ahora aplicamos la regla de la derivada de una divesión:

      Como resultado de la secuencia de reglas:

    Como resultado de la secuencia de reglas:

  4. Simplificamos:


Respuesta:

Primera derivada [src]
                                            ________       
                                           /    2          
                                          /    t           
    ________                             /   ------        
   /   1     /  t          t    \      \/    log(t)        
  /  ------ *|------ - ---------|*|t|*e             *log(t)
\/   log(t)  |log(t)        2   |                          
             \         2*log (t)/                          
-----------------------------------------------------------
                              2                            
                             t                             
$$\frac{\left(\frac{t}{\log{\left(t \right)}} - \frac{t}{2 \log{\left(t \right)}^{2}}\right) \sqrt{\frac{1}{\log{\left(t \right)}}} e^{\sqrt{\frac{t^{2}}{\log{\left(t \right)}}}} \log{\left(t \right)} \left|{t}\right|}{t^{2}}$$
Segunda derivada [src]
/                                                        ________                                                                                           \       ________
|            2       ________                           /   1     /      3         2   \           ________                        ________                 |      /    2   
|/      1   \       /   1     /      1   \             /  ------ *|2 - ------ + -------|*|t|      /   1     /      1   \          /   1     /      1   \    |     /    t    
||2 - ------|      /  ------ *|2 - ------|*sign(t)   \/   log(t)  |    log(t)      2   |         /  ------ *|2 - ------|*|t|     /  ------ *|2 - ------|*|t||    /   ------ 
|\    log(t)/    \/   log(t)  \    log(t)/                        \             log (t)/       \/   log(t)  \    log(t)/       \/   log(t)  \    log(t)/    |  \/    log(t) 
|------------- + --------------------------------- + --------------------------------------- - ----------------------------- + -----------------------------|*e             
|   4*log(t)                    2*t                                       2                                   2                            2                |               
\                                                                      2*t                                   t                          4*t *log(t)         /               
$$\left(\frac{\left(2 - \frac{1}{\log{\left(t \right)}}\right)^{2}}{4 \log{\left(t \right)}} + \frac{\left(2 - \frac{1}{\log{\left(t \right)}}\right) \sqrt{\frac{1}{\log{\left(t \right)}}} \operatorname{sign}{\left(t \right)}}{2 t} - \frac{\left(2 - \frac{1}{\log{\left(t \right)}}\right) \sqrt{\frac{1}{\log{\left(t \right)}}} \left|{t}\right|}{t^{2}} + \frac{\left(2 - \frac{1}{\log{\left(t \right)}}\right) \sqrt{\frac{1}{\log{\left(t \right)}}} \left|{t}\right|}{4 t^{2} \log{\left(t \right)}} + \frac{\left(2 - \frac{3}{\log{\left(t \right)}} + \frac{2}{\log{\left(t \right)}^{2}}\right) \sqrt{\frac{1}{\log{\left(t \right)}}} \left|{t}\right|}{2 t^{2}}\right) e^{\sqrt{\frac{t^{2}}{\log{\left(t \right)}}}}$$
Tercera derivada [src]
/                                                                                                                          ________                                                                              ________                                                                                                                                         ________                                  ________                                                                                                                         \       ________
|              2                 2                                               /      1   \ /      3         2   \      /   1     /      3         2   \                 ________                             /   1     /      3         2   \             ________                        ________             3           ________                           /   1     /      3         2   \          /   1     /      3         3   \             ________                        ________                                2            |      /    2   
|  /      1   \      /      1   \                                              3*|2 - ------|*|2 - ------ + -------|     /  ------ *|2 - ------ + -------|*sign(t)        /   1     /      1   \           2*  /  ------ *|2 - ------ + -------|*|t|        /   1     /      1   \          /   1     /      1   \           /   1     /      1   \             /  ------ *|2 - ------ + -------|*|t|     /  ------ *|1 - ------ + -------|*|t|        /   1     /      1   \          /   1     /      1   \       /      1   \             |     /    t    
|  |2 - ------|    3*|2 - ------|        ________                                \    log(t)/ |    log(t)      2   |   \/   log(t)  |    log(t)      2   |           2*  /  ------ *|2 - ------|*sign(t)     \/   log(t)  |    log(t)      2   |       3*  /  ------ *|2 - ------|*|t|     /  ------ *|2 - ------| *|t|     /  ------ *|2 - ------|*sign(t)   \/   log(t)  |    log(t)      2   |       \/   log(t)  |    log(t)      2   |       5*  /  ------ *|2 - ------|*|t|     /  ------ *|2 - ------|*|t|   |2 - ------| *|t|*sign(t)|    /   ------ 
|  \    log(t)/      \    log(t)/       /   1     /      1   \                                \             log (t)/                \             log (t)/             \/   log(t)  \    log(t)/                          \             log (t)/         \/   log(t)  \    log(t)/       \/   log(t)  \    log(t)/        \/   log(t)  \    log(t)/                        \             log (t)/                    \             log (t)/         \/   log(t)  \    log(t)/       \/   log(t)  \    log(t)/       \    log(t)/             |  \/    log(t) 
|- ------------- + --------------- +   /  ------ *|2 - ------|*DiracDelta(t) + ------------------------------------- + ------------------------------------------- - ----------------------------------- - ----------------------------------------- + ------------------------------- + ------------------------------ + --------------------------------- + --------------------------------------- - --------------------------------------- - ------------------------------- - ----------------------------- + -------------------------|*e             
|      log(t)              2         \/   log(t)  \    log(t)/                                4*log(t)                                      t                                         t                                         2                                      2                            8*log(t)                          2*t*log(t)                               2                                        2                                      2                                2    2                      4*t*log(t)       |               
\                     8*log (t)                                                                                                                                                                                                t                                      t                                                                                                     2*t *log(t)                                t *log(t)                            4*t *log(t)                      8*t *log (t)                                    /               
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                                                              t                                                                                                                                                                                                                                                                              
$$\frac{\left(\frac{\left(2 - \frac{1}{\log{\left(t \right)}}\right)^{3} \sqrt{\frac{1}{\log{\left(t \right)}}} \left|{t}\right|}{8 \log{\left(t \right)}} - \frac{\left(2 - \frac{1}{\log{\left(t \right)}}\right)^{2}}{\log{\left(t \right)}} + \frac{3 \left(2 - \frac{1}{\log{\left(t \right)}}\right)^{2}}{8 \log{\left(t \right)}^{2}} + \frac{3 \left(2 - \frac{1}{\log{\left(t \right)}}\right) \left(2 - \frac{3}{\log{\left(t \right)}} + \frac{2}{\log{\left(t \right)}^{2}}\right)}{4 \log{\left(t \right)}} + \left(2 - \frac{1}{\log{\left(t \right)}}\right) \sqrt{\frac{1}{\log{\left(t \right)}}} \delta\left(t\right) + \frac{\left(2 - \frac{1}{\log{\left(t \right)}}\right)^{2} \left|{t}\right| \operatorname{sign}{\left(t \right)}}{4 t \log{\left(t \right)}} - \frac{2 \left(2 - \frac{1}{\log{\left(t \right)}}\right) \sqrt{\frac{1}{\log{\left(t \right)}}} \operatorname{sign}{\left(t \right)}}{t} + \frac{\left(2 - \frac{1}{\log{\left(t \right)}}\right) \sqrt{\frac{1}{\log{\left(t \right)}}} \operatorname{sign}{\left(t \right)}}{2 t \log{\left(t \right)}} + \frac{\left(2 - \frac{3}{\log{\left(t \right)}} + \frac{2}{\log{\left(t \right)}^{2}}\right) \sqrt{\frac{1}{\log{\left(t \right)}}} \operatorname{sign}{\left(t \right)}}{t} + \frac{3 \left(2 - \frac{1}{\log{\left(t \right)}}\right) \sqrt{\frac{1}{\log{\left(t \right)}}} \left|{t}\right|}{t^{2}} - \frac{5 \left(2 - \frac{1}{\log{\left(t \right)}}\right) \sqrt{\frac{1}{\log{\left(t \right)}}} \left|{t}\right|}{4 t^{2} \log{\left(t \right)}} - \frac{\left(2 - \frac{1}{\log{\left(t \right)}}\right) \sqrt{\frac{1}{\log{\left(t \right)}}} \left|{t}\right|}{8 t^{2} \log{\left(t \right)}^{2}} - \frac{\left(1 - \frac{3}{\log{\left(t \right)}} + \frac{3}{\log{\left(t \right)}^{2}}\right) \sqrt{\frac{1}{\log{\left(t \right)}}} \left|{t}\right|}{t^{2} \log{\left(t \right)}} - \frac{2 \left(2 - \frac{3}{\log{\left(t \right)}} + \frac{2}{\log{\left(t \right)}^{2}}\right) \sqrt{\frac{1}{\log{\left(t \right)}}} \left|{t}\right|}{t^{2}} + \frac{\left(2 - \frac{3}{\log{\left(t \right)}} + \frac{2}{\log{\left(t \right)}^{2}}\right) \sqrt{\frac{1}{\log{\left(t \right)}}} \left|{t}\right|}{2 t^{2} \log{\left(t \right)}}\right) e^{\sqrt{\frac{t^{2}}{\log{\left(t \right)}}}}}{t}$$