Производная sin(factorial(x))

Вы ввели

sin(x!)

$$\sin{\left (x! \right )}$$

Первая производная

cos(x!)*gamma(1 + x)*polygamma(0, 1 + x)

$$\cos{\left (x! \right )} \Gamma{\left(x + 1 \right)} \operatorname{polygamma}{\left (0,x + 1 \right )}$$

Вторая производная

/         2                                                            2                               \             
\polygamma (0, 1 + x)*cos(x!) + cos(x!)*polygamma(1, 1 + x) - polygamma (0, 1 + x)*gamma(1 + x)*sin(x!)/*gamma(1 + x)

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

Третья производная

/         3                                                        2                 3                                3                                                                                                                                                    \             
\polygamma (0, 1 + x)*cos(x!) + cos(x!)*polygamma(2, 1 + x) - gamma (1 + x)*polygamma (0, 1 + x)*cos(x!) - 3*polygamma (0, 1 + x)*gamma(1 + x)*sin(x!) + 3*cos(x!)*polygamma(0, 1 + x)*polygamma(1, 1 + x) - 3*gamma(1 + x)*polygamma(0, 1 + x)*polygamma(1, 1 + x)*sin(x!)/*gamma(1 + x)

$$\left(- 3 \sin{\left (x! \right )} \Gamma{\left(x + 1 \right)} \operatorname{polygamma}^{3}{\left (0,x + 1 \right )} - 3 \sin{\left (x! \right )} \Gamma{\left(x + 1 \right)} \operatorname{polygamma}{\left (0,x + 1 \right )} \operatorname{polygamma}{\left (1,x + 1 \right )} - \cos{\left (x! \right )} \Gamma^{2}{\left(x + 1 \right)} \operatorname{polygamma}^{3}{\left (0,x + 1 \right )} + \cos{\left (x! \right )} \operatorname{polygamma}^{3}{\left (0,x + 1 \right )} + 3 \cos{\left (x! \right )} \operatorname{polygamma}{\left (0,x + 1 \right )} \operatorname{polygamma}{\left (1,x + 1 \right )} + \cos{\left (x! \right )} \operatorname{polygamma}{\left (2,x + 1 \right )}\right) \Gamma{\left(x + 1 \right)}$$