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

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

Gráfico:

interior superior

Definida a trozos:

Solución

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