Производная (factorial(x))^(1/x)

Вы ввели

x ____
\/ x! 

$$x!^{\frac{1}{x}}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

   $\left(\log{\left (\frac{1}{x} \right )} + 1\right) \left(\frac{1}{x}\right)^{\frac{1}{x}}$

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Ответ:

$\left(\log{\left (\frac{1}{x} \right )} + 1\right) \left(\frac{1}{x}\right)^{\frac{1}{x}}$

Первая производная

x ____ /  log(x!)   gamma(1 + x)*polygamma(0, 1 + x)\
\/ x! *|- ------- + --------------------------------|
       |      2                   x*x!              |
       \     x                                      /

$$\left(\frac{\Gamma{\left(x + 1 \right)}}{x x!} \operatorname{polygamma}{\left (0,x + 1 \right )} - \frac{1}{x^{2}} \log{\left (x! \right )}\right) x!^{\frac{1}{x}}$$

Вторая производная

       /                                              2                                                                                                                                                             \
       |/  log(x!)   gamma(1 + x)*polygamma(0, 1 + x)\                                                                                                                                                              |
       ||- ------- + --------------------------------|                         2                                                                  2                 2                                               |
x ____ |\     x                     x!               /    2*log(x!)   polygamma (0, 1 + x)*gamma(1 + x)   gamma(1 + x)*polygamma(1, 1 + x)   gamma (1 + x)*polygamma (0, 1 + x)   2*gamma(1 + x)*polygamma(0, 1 + x)|
\/ x! *|----------------------------------------------- + --------- + --------------------------------- + -------------------------------- - ---------------------------------- - ----------------------------------|
       |                       x                               2                      x!                                 x!                                   2                                  x*x!               |
       \                                                      x                                                                                             x!                                                      /
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$$\frac{x!^{\frac{1}{x}}}{x} \left(\frac{1}{x!} \Gamma{\left(x + 1 \right)} \operatorname{polygamma}^{2}{\left (0,x + 1 \right )} + \frac{1}{x!} \Gamma{\left(x + 1 \right)} \operatorname{polygamma}{\left (1,x + 1 \right )} - \frac{1}{x!^{2}} \Gamma^{2}{\left(x + 1 \right)} \operatorname{polygamma}^{2}{\left (0,x + 1 \right )} + \frac{1}{x} \left(\frac{1}{x!} \Gamma{\left(x + 1 \right)} \operatorname{polygamma}{\left (0,x + 1 \right )} - \frac{1}{x} \log{\left (x! \right )}\right)^{2} - \frac{2}{x x!} \Gamma{\left(x + 1 \right)} \operatorname{polygamma}{\left (0,x + 1 \right )} + \frac{2}{x^{2}} \log{\left (x! \right )}\right)$$

Третья производная

       /                                                                                                                                                                                                                                                                    /                     2                                                                  2                 2                                               \                                                                                                                                                                                                                                                                          \
       |                                              3                                                                                                                                                                      /  log(x!)   gamma(1 + x)*polygamma(0, 1 + x)\ |2*log(x!)   polygamma (0, 1 + x)*gamma(1 + x)   gamma(1 + x)*polygamma(1, 1 + x)   gamma (1 + x)*polygamma (0, 1 + x)   2*gamma(1 + x)*polygamma(0, 1 + x)|                                                                                                                                                                                                                                                                          |
       |/  log(x!)   gamma(1 + x)*polygamma(0, 1 + x)\                                                                                                                                                                     3*|- ------- + --------------------------------|*|--------- + --------------------------------- + -------------------------------- - ---------------------------------- - ----------------------------------|                                                                                                                                                                                                                                                                          |
       ||- ------- + --------------------------------|                         3                                                                    2                 3                    3                 3               \     x                     x!               / |     2                      x!                                 x!                                   2                                  x*x!               |              2                                                                      2                                                         2                 2                                                                                                        |
x ____ |\     x                     x!               /    6*log(x!)   polygamma (0, 1 + x)*gamma(1 + x)   gamma(1 + x)*polygamma(2, 1 + x)   3*gamma (1 + x)*polygamma (0, 1 + x)   2*gamma (1 + x)*polygamma (0, 1 + x)                                                    \    x                                                                                             x!                                                      /   3*polygamma (0, 1 + x)*gamma(1 + x)   3*gamma(1 + x)*polygamma(1, 1 + x)   3*gamma (1 + x)*polygamma(0, 1 + x)*polygamma(1, 1 + x)   3*gamma (1 + x)*polygamma (0, 1 + x)   3*gamma(1 + x)*polygamma(0, 1 + x)*polygamma(1, 1 + x)   6*gamma(1 + x)*polygamma(0, 1 + x)|
\/ x! *|----------------------------------------------- - --------- + --------------------------------- + -------------------------------- - ------------------------------------ + ------------------------------------ + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------- - ---------------------------------- - ------------------------------------------------------- + ------------------------------------ + ------------------------------------------------------ + ----------------------------------|
       |                        2                              3                      x!                                 x!                                    2                                      3                                                                                                                          x                                                                                                                         x*x!                                 x*x!                                              2                                                2                                             x!                                            2                  |
       \                       x                              x                                                                                              x!                                     x!                                                                                                                                                                                                                                                                                                                                          x!                                             x*x!                                                                                           x *x!               /
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$$\frac{x!^{\frac{1}{x}}}{x} \left(\frac{1}{x!} \Gamma{\left(x + 1 \right)} \operatorname{polygamma}^{3}{\left (0,x + 1 \right )} + \frac{3}{x!} \Gamma{\left(x + 1 \right)} \operatorname{polygamma}{\left (0,x + 1 \right )} \operatorname{polygamma}{\left (1,x + 1 \right )} + \frac{1}{x!} \Gamma{\left(x + 1 \right)} \operatorname{polygamma}{\left (2,x + 1 \right )} - \frac{3}{x!^{2}} \Gamma^{2}{\left(x + 1 \right)} \operatorname{polygamma}^{3}{\left (0,x + 1 \right )} - \frac{3}{x!^{2}} \Gamma^{2}{\left(x + 1 \right)} \operatorname{polygamma}{\left (0,x + 1 \right )} \operatorname{polygamma}{\left (1,x + 1 \right )} + \frac{2}{x!^{3}} \Gamma^{3}{\left(x + 1 \right)} \operatorname{polygamma}^{3}{\left (0,x + 1 \right )} + \frac{3}{x} \left(\frac{1}{x!} \Gamma{\left(x + 1 \right)} \operatorname{polygamma}{\left (0,x + 1 \right )} - \frac{1}{x} \log{\left (x! \right )}\right) \left(\frac{1}{x!} \Gamma{\left(x + 1 \right)} \operatorname{polygamma}^{2}{\left (0,x + 1 \right )} + \frac{1}{x!} \Gamma{\left(x + 1 \right)} \operatorname{polygamma}{\left (1,x + 1 \right )} - \frac{1}{x!^{2}} \Gamma^{2}{\left(x + 1 \right)} \operatorname{polygamma}^{2}{\left (0,x + 1 \right )} - \frac{2}{x x!} \Gamma{\left(x + 1 \right)} \operatorname{polygamma}{\left (0,x + 1 \right )} + \frac{2}{x^{2}} \log{\left (x! \right )}\right) - \frac{3}{x x!} \Gamma{\left(x + 1 \right)} \operatorname{polygamma}^{2}{\left (0,x + 1 \right )} - \frac{3}{x x!} \Gamma{\left(x + 1 \right)} \operatorname{polygamma}{\left (1,x + 1 \right )} + \frac{3}{x x!^{2}} \Gamma^{2}{\left(x + 1 \right)} \operatorname{polygamma}^{2}{\left (0,x + 1 \right )} + \frac{1}{x^{2}} \left(\frac{1}{x!} \Gamma{\left(x + 1 \right)} \operatorname{polygamma}{\left (0,x + 1 \right )} - \frac{1}{x} \log{\left (x! \right )}\right)^{3} + \frac{6}{x^{2} x!} \Gamma{\left(x + 1 \right)} \operatorname{polygamma}{\left (0,x + 1 \right )} - \frac{6}{x^{3}} \log{\left (x! \right )}\right)$$