Господин Экзамен
Производная atan(x)/((|x|-1))
Решение
Вы ввели
[src]
atan(x) ------- |x| - 1
$$\frac{\operatorname{atan}{\left(x \right)}}{\left|{x}\right| - 1}$$
d /atan(x)\ --|-------| dx\|x| - 1/
$$\frac{d}{d x} \frac{\operatorname{atan}{\left(x \right)}}{\left|{x}\right| - 1}$$
Первая производная
[src]
1 atan(x)*sign(x) ------------------ - --------------- / 2\ 2 \1 + x /*(|x| - 1) (|x| - 1)
$$- \frac{\operatorname{atan}{\left(x \right)} \operatorname{sign}{\left(x \right)}}{\left(\left|{x}\right| - 1\right)^{2}} + \frac{1}{\left(x^{2} + 1\right) \left(\left|{x}\right| - 1\right)}$$
Вторая производная
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/ / 2 \ \
| | sign (x) | |
| |- -------- + DiracDelta(x)|*atan(x)|
| x sign(x) \ -1 + |x| / |
-2*|--------- + ------------------- + ------------------------------------|
| 2 / 2\ -1 + |x| |
|/ 2\ \1 + x /*(-1 + |x|) |
\\1 + x / /
---------------------------------------------------------------------------
-1 + |x|
$$- \frac{2 \left(\frac{\left(\delta\left(x\right) - \frac{\operatorname{sign}^{2}{\left(x \right)}}{\left|{x}\right| - 1}\right) \operatorname{atan}{\left(x \right)}}{\left|{x}\right| - 1} + \frac{x}{\left(x^{2} + 1\right)^{2}} + \frac{\operatorname{sign}{\left(x \right)}}{\left(x^{2} + 1\right) \left(\left|{x}\right| - 1\right)}\right)}{\left|{x}\right| - 1}$$
Третья производная
[src]
/ 2 / 3 \ \
| 4*x | 3*sign (x) 6*DiracDelta(x)*sign(x) | / 2 \ |
|-1 + ------ |----------- - ----------------------- + DiracDelta(x, 1)|*atan(x) | sign (x) | |
| 2 | 2 -1 + |x| | 3*|- -------- + DiracDelta(x)| |
| 1 + x \(-1 + |x|) / \ -1 + |x| / 3*x*sign(x) |
2*|----------- - ------------------------------------------------------------------ - ------------------------------ + --------------------|
| 2 -1 + |x| / 2\ 2 |
| / 2\ \1 + x /*(-1 + |x|) / 2\ |
\ \1 + x / \1 + x / *(-1 + |x|)/
--------------------------------------------------------------------------------------------------------------------------------------------
-1 + |x|
$$\frac{2 \cdot \left(- \frac{\left(\delta^{\left( 1 \right)}\left( x \right) - \frac{6 \delta\left(x\right) \operatorname{sign}{\left(x \right)}}{\left|{x}\right| - 1} + \frac{3 \operatorname{sign}^{3}{\left(x \right)}}{\left(\left|{x}\right| - 1\right)^{2}}\right) \operatorname{atan}{\left(x \right)}}{\left|{x}\right| - 1} + \frac{3 x \operatorname{sign}{\left(x \right)}}{\left(x^{2} + 1\right)^{2} \left(\left|{x}\right| - 1\right)} - \frac{3 \left(\delta\left(x\right) - \frac{\operatorname{sign}^{2}{\left(x \right)}}{\left|{x}\right| - 1}\right)}{\left(x^{2} + 1\right) \left(\left|{x}\right| - 1\right)} + \frac{\frac{4 x^{2}}{x^{2} + 1} - 1}{\left(x^{2} + 1\right)^{2}}\right)}{\left|{x}\right| - 1}$$