Господин Экзамен
Вы ввели:
4^n/factorial(n-1)
Что Вы имели ввиду?
Производная 4^n/factorial(n-1)
Решение
Вы ввели
[src]
n 4 -------- (n - 1)!
$$\frac{4^{n}}{\left(n - 1\right)!}$$
/ n \ d | 4 | --|--------| dn\(n - 1)!/
$$\frac{d}{d n} \frac{4^{n}}{\left(n - 1\right)!}$$
Первая производная
[src]
n n
4 *log(4) 4 *Gamma(1 - 1 + n)*polygamma(0, n)
--------- - -----------------------------------
(n - 1)! 2
(n - 1)!
$$- \frac{4^{n} \Gamma\left(n - 1 + 1\right) \operatorname{polygamma}{\left(0,n \right)}}{\left(n - 1\right)!^{2}} + \frac{4^{n} \log{\left(4 \right)}}{\left(n - 1\right)!}$$
Вторая производная
[src]
/ / 2 \ \
| | 2 2*polygamma (0, n)*Gamma(n) | |
| |polygamma (0, n) - --------------------------- + polygamma(1, n)|*Gamma(n) |
n | 2 \ (-1 + n)! / 2*Gamma(n)*log(4)*polygamma(0, n)|
4 *|log (4) - --------------------------------------------------------------------------- - ---------------------------------|
\ (-1 + n)! (-1 + n)! /
------------------------------------------------------------------------------------------------------------------------------
(-1 + n)!
$$\frac{4^{n} \left(- \frac{\left(\operatorname{polygamma}^{2}{\left(0,n \right)} - \frac{2 \Gamma\left(n\right) \operatorname{polygamma}^{2}{\left(0,n \right)}}{\left(n - 1\right)!} + \operatorname{polygamma}{\left(1,n \right)}\right) \Gamma\left(n\right)}{\left(n - 1\right)!} - \frac{2 \log{\left(4 \right)} \Gamma\left(n\right) \operatorname{polygamma}{\left(0,n \right)}}{\left(n - 1\right)!} + \log{\left(4 \right)}^{2}\right)}{\left(n - 1\right)!}$$
Третья производная
[src]
/ / 3 2 3 \ \
| | 3 6*polygamma (0, n)*Gamma(n) 6*Gamma (n)*polygamma (0, n) 6*Gamma(n)*polygamma(0, n)*polygamma(1, n) | / 2 \ |
| |polygamma (0, n) + 3*polygamma(0, n)*polygamma(1, n) - --------------------------- + ---------------------------- - ------------------------------------------ + polygamma(2, n)|*Gamma(n) | 2 2*polygamma (0, n)*Gamma(n) | |
| | (-1 + n)! 2 (-1 + n)! | 2 3*|polygamma (0, n) - --------------------------- + polygamma(1, n)|*Gamma(n)*log(4)|
n | 3 \ (-1 + n)! / 3*log (4)*Gamma(n)*polygamma(0, n) \ (-1 + n)! / |
4 *|log (4) - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------- - ------------------------------------------------------------------------------------|
\ (-1 + n)! (-1 + n)! (-1 + n)! /
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
(-1 + n)!
$$\frac{4^{n} \left(- \frac{3 \left(\operatorname{polygamma}^{2}{\left(0,n \right)} - \frac{2 \Gamma\left(n\right) \operatorname{polygamma}^{2}{\left(0,n \right)}}{\left(n - 1\right)!} + \operatorname{polygamma}{\left(1,n \right)}\right) \log{\left(4 \right)} \Gamma\left(n\right)}{\left(n - 1\right)!} - \frac{\left(\operatorname{polygamma}^{3}{\left(0,n \right)} - \frac{6 \Gamma\left(n\right) \operatorname{polygamma}^{3}{\left(0,n \right)}}{\left(n - 1\right)!} + \frac{6 \Gamma^{2}\left(n\right) \operatorname{polygamma}^{3}{\left(0,n \right)}}{\left(n - 1\right)!^{2}} + 3 \operatorname{polygamma}{\left(0,n \right)} \operatorname{polygamma}{\left(1,n \right)} - \frac{6 \Gamma\left(n\right) \operatorname{polygamma}{\left(0,n \right)} \operatorname{polygamma}{\left(1,n \right)}}{\left(n - 1\right)!} + \operatorname{polygamma}{\left(2,n \right)}\right) \Gamma\left(n\right)}{\left(n - 1\right)!} - \frac{3 \log{\left(4 \right)}^{2} \Gamma\left(n\right) \operatorname{polygamma}{\left(0,n \right)}}{\left(n - 1\right)!} + \log{\left(4 \right)}^{3}\right)}{\left(n - 1\right)!}$$