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Derivada de ((x^k))*(exp^(-k))/k!

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

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src]
 k  -k
x *E  
------
  k!  
$$\frac{e^{- k} x^{k}}{k!}$$
(x^k*E^(-k))/factorial(k)
Primera derivada [src]
   k  -k    k  -k           k  -k                                 
- x *e   + x *e  *log(x)   x *e  *Gamma(1 + k)*polygamma(0, 1 + k)
------------------------ - ---------------------------------------
           k!                                  2                  
                                             k!                   
$$- \frac{x^{k} e^{- k} \Gamma\left(k + 1\right) \operatorname{polygamma}{\left(0,k + 1 \right)}}{k!^{2}} + \frac{x^{k} e^{- k} \log{\left(x \right)} - x^{k} e^{- k}}{k!}$$
Segunda derivada [src]
   /                         /                                  2                                             \                                                                \    
   |                         |         2             2*polygamma (0, 1 + k)*Gamma(1 + k)                      |                                                                |    
   |                         |polygamma (0, 1 + k) - ----------------------------------- + polygamma(1, 1 + k)|*Gamma(1 + k)                                                   |    
 k |       2                 \                                        k!                                      /                2*(-1 + log(x))*Gamma(1 + k)*polygamma(0, 1 + k)|  -k
x *|1 + log (x) - 2*log(x) - ----------------------------------------------------------------------------------------------- - ------------------------------------------------|*e  
   \                                                                        k!                                                                        k!                       /    
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                         k!                                                                                         
$$\frac{x^{k} \left(- \frac{2 \left(\log{\left(x \right)} - 1\right) \Gamma\left(k + 1\right) \operatorname{polygamma}{\left(0,k + 1 \right)}}{k!} - \frac{\left(\operatorname{polygamma}^{2}{\left(0,k + 1 \right)} + \operatorname{polygamma}{\left(1,k + 1 \right)} - \frac{2 \Gamma\left(k + 1\right) \operatorname{polygamma}^{2}{\left(0,k + 1 \right)}}{k!}\right) \Gamma\left(k + 1\right)}{k!} + \log{\left(x \right)}^{2} - 2 \log{\left(x \right)} + 1\right) e^{- k}}{k!}$$
Tercera derivada [src]
   /                                      /                                                                              3                                 2                 3                                                                                         \                                                                                                                                                                                             \    
   |                                      |         3                                                         6*polygamma (0, 1 + k)*Gamma(1 + k)   6*Gamma (1 + k)*polygamma (0, 1 + k)   6*Gamma(1 + k)*polygamma(0, 1 + k)*polygamma(1, 1 + k)                      |                                /                                  2                                             \                                                                           |    
   |                                      |polygamma (0, 1 + k) + 3*polygamma(0, 1 + k)*polygamma(1, 1 + k) - ----------------------------------- + ------------------------------------ - ------------------------------------------------------ + polygamma(2, 1 + k)|*Gamma(1 + k)                   |         2             2*polygamma (0, 1 + k)*Gamma(1 + k)                      |                                                                           |    
   |                                      |                                                                                    k!                                     2                                              k!                                                |                3*(-1 + log(x))*|polygamma (0, 1 + k) - ----------------------------------- + polygamma(1, 1 + k)|*Gamma(1 + k)     /       2              \                                 |    
 k |        3           2                 \                                                                                                                         k!                                                                                                 /                                \                                        k!                                      /                3*\1 + log (x) - 2*log(x)/*Gamma(1 + k)*polygamma(0, 1 + k)|  -k
x *|-1 + log (x) - 3*log (x) + 3*log(x) - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------- - -----------------------------------------------------------|*e  
   \                                                                                                                                                           k!                                                                                                                                                                              k!                                                                                      k!                            /    
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                    k!                                                                                                                                                                                                                                    
$$\frac{x^{k} \left(- \frac{3 \left(\log{\left(x \right)} - 1\right) \left(\operatorname{polygamma}^{2}{\left(0,k + 1 \right)} + \operatorname{polygamma}{\left(1,k + 1 \right)} - \frac{2 \Gamma\left(k + 1\right) \operatorname{polygamma}^{2}{\left(0,k + 1 \right)}}{k!}\right) \Gamma\left(k + 1\right)}{k!} - \frac{3 \left(\log{\left(x \right)}^{2} - 2 \log{\left(x \right)} + 1\right) \Gamma\left(k + 1\right) \operatorname{polygamma}{\left(0,k + 1 \right)}}{k!} - \frac{\left(\operatorname{polygamma}^{3}{\left(0,k + 1 \right)} + 3 \operatorname{polygamma}{\left(0,k + 1 \right)} \operatorname{polygamma}{\left(1,k + 1 \right)} + \operatorname{polygamma}{\left(2,k + 1 \right)} - \frac{6 \Gamma\left(k + 1\right) \operatorname{polygamma}^{3}{\left(0,k + 1 \right)}}{k!} - \frac{6 \Gamma\left(k + 1\right) \operatorname{polygamma}{\left(0,k + 1 \right)} \operatorname{polygamma}{\left(1,k + 1 \right)}}{k!} + \frac{6 \Gamma^{2}\left(k + 1\right) \operatorname{polygamma}^{3}{\left(0,k + 1 \right)}}{k!^{2}}\right) \Gamma\left(k + 1\right)}{k!} + \log{\left(x \right)}^{3} - 3 \log{\left(x \right)}^{2} + 3 \log{\left(x \right)} - 1\right) e^{- k}}{k!}$$