Господин Экзамен
Производная x/(|x|+1)
Решение
Вы ввели
[src]
x ------- |x| + 1
$$\frac{x}{\left|{x}\right| + 1}$$
d / x \ --|-------| dx\|x| + 1/
$$\frac{d}{d x} \frac{x}{\left|{x}\right| + 1}$$
Первая производная
[src]
1 x*sign(x)
------- - ----------
|x| + 1 2
(|x| + 1)
$$- \frac{x \operatorname{sign}{\left(x \right)}}{\left(\left|{x}\right| + 1\right)^{2}} + \frac{1}{\left|{x}\right| + 1}$$
Вторая производная
[src]
/ / 2 \ \
| | sign (x) | |
-2*|x*|- -------- + DiracDelta(x)| + sign(x)|
\ \ 1 + |x| / /
---------------------------------------------
2
(1 + |x|)
$$- \frac{2 \left(x \left(\delta\left(x\right) - \frac{\operatorname{sign}^{2}{\left(x \right)}}{\left|{x}\right| + 1}\right) + \operatorname{sign}{\left(x \right)}\right)}{\left(\left|{x}\right| + 1\right)^{2}}$$
Третья производная
[src]
/ / 3 \ 2 \
| |3*sign (x) 6*DiracDelta(x)*sign(x) | 3*sign (x)|
-2*|3*DiracDelta(x) + x*|---------- - ----------------------- + DiracDelta(x, 1)| - ----------|
| | 2 1 + |x| | 1 + |x| |
\ \(1 + |x|) / /
-----------------------------------------------------------------------------------------------
2
(1 + |x|)
$$- \frac{2 \left(x \left(\delta^{\left( 1 \right)}\left( x \right) - \frac{6 \delta\left(x\right) \operatorname{sign}{\left(x \right)}}{\left|{x}\right| + 1} + \frac{3 \operatorname{sign}^{3}{\left(x \right)}}{\left(\left|{x}\right| + 1\right)^{2}}\right) + 3 \delta\left(x\right) - \frac{3 \operatorname{sign}^{2}{\left(x \right)}}{\left|{x}\right| + 1}\right)}{\left(\left|{x}\right| + 1\right)^{2}}$$