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Derivada de la función/ ((x+m-1)!/x!)q^x

Derivada de ((x+m-1)!/x!)q^x

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

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
(x + m - 1)!  x
------------*q 
     x!
qx((m+x)−1)!x!q^{x} \frac{\left(\left(m + x\right) - 1\right)!}{x!} 
(factorial(x + m - 1)/factorial(x))*q^x
Primera derivada [src] 
                                                                                         x                    
 x /Gamma(m + x)*polygamma(0, m + x)   (x + m - 1)!*Gamma(1 + x)*polygamma(0, 1 + x)\   q *(x + m - 1)!*log(q)
q *|-------------------------------- - ---------------------------------------------| + ----------------------
   |               x!                                         2                     |             x!          
   \                                                        x!                      /
qx(Γ(m+x)polygamma⁡(0,m+x)x!−((m+x)−1)!Γ(x+1)polygamma⁡(0,x+1)x!2)+qxlog⁡(q)((m+x)−1)!x!q^{x} \left(\frac{\Gamma\left(m + x\right) \operatorname{polygamma}{\left(0,m + x \right)}}{x!} - \frac{\left(\left(m + x\right) - 1\right)! \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{x!^{2}}\right) + \frac{q^{x} \log{\left(q \right)} \left(\left(m + x\right) - 1\right)!}{x!}
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Segunda derivada [src] 
   /                                                                                                                                                                                   /                                  2                                             \                                                                                                 \
   |                                                                                                                                                                                   |         2             2*polygamma (0, 1 + x)*Gamma(1 + x)                      |                                                                                                 |
   |                                                                                                                                                                                   |polygamma (0, 1 + x) - ----------------------------------- + polygamma(1, 1 + x)|*(-1 + m + x)!*Gamma(1 + x)                                                                      |
 x |   2                    /         2                                \                  /                                   (-1 + m + x)!*Gamma(1 + x)*polygamma(0, 1 + x)\          \                                        x!                                      /                              2*Gamma(1 + x)*Gamma(m + x)*polygamma(0, 1 + x)*polygamma(0, m + x)|
q *|log (q)*(-1 + m + x)! + \polygamma (0, m + x) + polygamma(1, m + x)/*Gamma(m + x) + 2*|Gamma(m + x)*polygamma(0, m + x) - ----------------------------------------------|*log(q) - ------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------|
   \                                                                                      \                                                         x!                      /                                                                x!                                                                                         x!                                /
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                     x!
qx(2(Γ(m+x)polygamma⁡(0,m+x)−(m+x−1)!Γ(x+1)polygamma⁡(0,x+1)x!)log⁡(q)+(polygamma⁡2(0,m+x)+polygamma⁡(1,m+x))Γ(m+x)−(polygamma⁡2(0,x+1)+polygamma⁡(1,x+1)−2Γ(x+1)polygamma⁡2(0,x+1)x!)(m+x−1)!Γ(x+1)x!+log⁡(q)2(m+x−1)!−2Γ(m+x)Γ(x+1)polygamma⁡(0,m+x)polygamma⁡(0,x+1)x!)x!\frac{q^{x} \left(2 \left(\Gamma\left(m + x\right) \operatorname{polygamma}{\left(0,m + x \right)} - \frac{\left(m + x - 1\right)! \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{x!}\right) \log{\left(q \right)} + \left(\operatorname{polygamma}^{2}{\left(0,m + x \right)} + \operatorname{polygamma}{\left(1,m + x \right)}\right) \Gamma\left(m + x\right) - \frac{\left(\operatorname{polygamma}^{2}{\left(0,x + 1 \right)} + \operatorname{polygamma}{\left(1,x + 1 \right)} - \frac{2 \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{x!}\right) \left(m + x - 1\right)! \Gamma\left(x + 1\right)}{x!} + \log{\left(q \right)}^{2} \left(m + x - 1\right)! - \frac{2 \Gamma\left(m + x\right) \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,m + x \right)} \operatorname{polygamma}{\left(0,x + 1 \right)}}{x!}\right)}{x!}
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Tercera derivada [src] 
   /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               /                                                                              3                                 2                 3                                                                                         \                                                                                                                                                                                                                                                               \
   |                                                                                                                                  /                                                              /                                  2                                             \                                                                                                 \                                                                                                          |         3                                                         6*polygamma (0, 1 + x)*Gamma(1 + x)   6*Gamma (1 + x)*polygamma (0, 1 + x)   6*Gamma(1 + x)*polygamma(0, 1 + x)*polygamma(1, 1 + x)                      |                                                                                                                               /                                  2                                             \                                              |
   |                                                                                                                                  |                                                              |         2             2*polygamma (0, 1 + x)*Gamma(1 + x)                      |                                                                                                 |                                                                                                          |polygamma (0, 1 + x) + 3*polygamma(0, 1 + x)*polygamma(1, 1 + x) - ----------------------------------- + ------------------------------------ - ------------------------------------------------------ + polygamma(2, 1 + x)|*(-1 + m + x)!*Gamma(1 + x)                                                                                                    |         2             2*polygamma (0, 1 + x)*Gamma(1 + x)                      |                                              |
   |                                                                                                                                  |                                                              |polygamma (0, 1 + x) - ----------------------------------- + polygamma(1, 1 + x)|*(-1 + m + x)!*Gamma(1 + x)                                                                      |                                                                                                          |                                                                                    x!                                     2                                              x!                                                |                                /         2                                \                                                 3*|polygamma (0, 1 + x) - ----------------------------------- + polygamma(1, 1 + x)|*Gamma(1 + x)*Gamma(m + x)*polygamma(0, m + x)|
 x |   3                    /         3                                                                            \                  |  /         2                                \                \                                        x!                                      /                              2*Gamma(1 + x)*Gamma(m + x)*polygamma(0, 1 + x)*polygamma(0, m + x)|               2    /                                   (-1 + m + x)!*Gamma(1 + x)*polygamma(0, 1 + x)\   \                                                                                                                         x!                                                                                                 /                              3*\polygamma (0, m + x) + polygamma(1, m + x)/*Gamma(1 + x)*Gamma(m + x)*polygamma(0, 1 + x)     \                                        x!                                      /                                              |
q *|log (q)*(-1 + m + x)! + \polygamma (0, m + x) + 3*polygamma(0, m + x)*polygamma(1, m + x) + polygamma(2, m + x)/*Gamma(m + x) - 3*|- \polygamma (0, m + x) + polygamma(1, m + x)/*Gamma(m + x) + ------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------|*log(q) + 3*log (q)*|Gamma(m + x)*polygamma(0, m + x) - ----------------------------------------------| - --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------|
   \                                                                                                                                  \                                                                                                                    x!                                                                                         x!                                /                    \                                                         x!                      /                                                                                                                               x!                                                                                                                                                                           x!                                                                                                                x!                                                                /
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                x!
qx(3(Γ(m+x)polygamma⁡(0,m+x)−(m+x−1)!Γ(x+1)polygamma⁡(0,x+1)x!)log⁡(q)2−3(polygamma⁡2(0,m+x)+polygamma⁡(1,m+x))Γ(m+x)Γ(x+1)polygamma⁡(0,x+1)x!−3(−(polygamma⁡2(0,m+x)+polygamma⁡(1,m+x))Γ(m+x)+(polygamma⁡2(0,x+1)+polygamma⁡(1,x+1)−2Γ(x+1)polygamma⁡2(0,x+1)x!)(m+x−1)!Γ(x+1)x!+2Γ(m+x)Γ(x+1)polygamma⁡(0,m+x)polygamma⁡(0,x+1)x!)log⁡(q)+(polygamma⁡3(0,m+x)+3polygamma⁡(0,m+x)polygamma⁡(1,m+x)+polygamma⁡(2,m+x))Γ(m+x)−3(polygamma⁡2(0,x+1)+polygamma⁡(1,x+1)−2Γ(x+1)polygamma⁡2(0,x+1)x!)Γ(m+x)Γ(x+1)polygamma⁡(0,m+x)x!−(polygamma⁡3(0,x+1)+3polygamma⁡(0,x+1)polygamma⁡(1,x+1)+polygamma⁡(2,x+1)−6Γ(x+1)polygamma⁡3(0,x+1)x!−6Γ(x+1)polygamma⁡(0,x+1)polygamma⁡(1,x+1)x!+6Γ2(x+1)polygamma⁡3(0,x+1)x!2)(m+x−1)!Γ(x+1)x!+log⁡(q)3(m+x−1)!)x!\frac{q^{x} \left(3 \left(\Gamma\left(m + x\right) \operatorname{polygamma}{\left(0,m + x \right)} - \frac{\left(m + x - 1\right)! \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{x!}\right) \log{\left(q \right)}^{2} - \frac{3 \left(\operatorname{polygamma}^{2}{\left(0,m + x \right)} + \operatorname{polygamma}{\left(1,m + x \right)}\right) \Gamma\left(m + x\right) \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{x!} - 3 \left(- \left(\operatorname{polygamma}^{2}{\left(0,m + x \right)} + \operatorname{polygamma}{\left(1,m + x \right)}\right) \Gamma\left(m + x\right) + \frac{\left(\operatorname{polygamma}^{2}{\left(0,x + 1 \right)} + \operatorname{polygamma}{\left(1,x + 1 \right)} - \frac{2 \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{x!}\right) \left(m + x - 1\right)! \Gamma\left(x + 1\right)}{x!} + \frac{2 \Gamma\left(m + x\right) \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,m + x \right)} \operatorname{polygamma}{\left(0,x + 1 \right)}}{x!}\right) \log{\left(q \right)} + \left(\operatorname{polygamma}^{3}{\left(0,m + x \right)} + 3 \operatorname{polygamma}{\left(0,m + x \right)} \operatorname{polygamma}{\left(1,m + x \right)} + \operatorname{polygamma}{\left(2,m + x \right)}\right) \Gamma\left(m + x\right) - \frac{3 \left(\operatorname{polygamma}^{2}{\left(0,x + 1 \right)} + \operatorname{polygamma}{\left(1,x + 1 \right)} - \frac{2 \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{x!}\right) \Gamma\left(m + x\right) \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,m + x \right)}}{x!} - \frac{\left(\operatorname{polygamma}^{3}{\left(0,x + 1 \right)} + 3 \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} + \operatorname{polygamma}{\left(2,x + 1 \right)} - \frac{6 \Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)}}{x!} - \frac{6 \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)}}{x!} + \frac{6 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)}}{x!^{2}}\right) \left(m + x - 1\right)! \Gamma\left(x + 1\right)}{x!} + \log{\left(q \right)}^{3} \left(m + x - 1\right)!\right)}{x!}
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