Производная atan(x+1)+(x+1)/(x^2+2*x+2)
Решение
Вы ввели
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x + 1
atan(x + 1) + ------------
2
x + 2*x + 2
$$\frac{x + 1}{x^{2} + 2 x + 2} + \operatorname{atan}{\left (x + 1 \right )}$$
График
Первая производная
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1 1 (-2 - 2*x)*(x + 1)
------------ + ------------ + ------------------
2 2 2
1 + (x + 1) x + 2*x + 2 / 2 \
\x + 2*x + 2/
$$\frac{\left(- 2 x - 2\right) \left(x + 1\right)}{\left(x^{2} + 2 x + 2\right)^{2}} + \frac{1}{x^{2} + 2 x + 2} + \frac{1}{\left(x + 1\right)^{2} + 1}$$
Вторая производная
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/ 2 \
| 1 3 4*(1 + x) |
2*(1 + x)*|- --------------- - --------------- + ---------------|
| 2 2 3|
| / 2\ / 2 \ / 2 \ |
\ \1 + (1 + x) / \2 + x + 2*x/ \2 + x + 2*x/ /
$$2 \left(x + 1\right) \left(\frac{4 \left(x + 1\right)^{2}}{\left(x^{2} + 2 x + 2\right)^{3}} - \frac{3}{\left(x^{2} + 2 x + 2\right)^{2}} - \frac{1}{\left(\left(x + 1\right)^{2} + 1\right)^{2}}\right)$$
Третья производная
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/ 4 2 2 \ | 1 3 24*(1 + x) 4*(1 + x) 24*(1 + x) | 2*|- --------------- - --------------- - --------------- + --------------- + ---------------| | 2 2 4 3 3| | / 2\ / 2 \ / 2 \ / 2\ / 2 \ | \ \1 + (1 + x) / \2 + x + 2*x/ \2 + x + 2*x/ \1 + (1 + x) / \2 + x + 2*x/ /
$$2 \left(- \frac{24 \left(x + 1\right)^{4}}{\left(x^{2} + 2 x + 2\right)^{4}} + \frac{24 \left(x + 1\right)^{2}}{\left(x^{2} + 2 x + 2\right)^{3}} + \frac{4 \left(x + 1\right)^{2}}{\left(\left(x + 1\right)^{2} + 1\right)^{3}} - \frac{3}{\left(x^{2} + 2 x + 2\right)^{2}} - \frac{1}{\left(\left(x + 1\right)^{2} + 1\right)^{2}}\right)$$