Производная (cot(x)^7)^atan(sqrt(x))
Решение
Вы ввели
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/ ___\
atan\\/ x /
7
cot (x)
$$\left(\cot^{7}{\left (x \right )}\right)^{\operatorname{atan}{\left (\sqrt{x} \right )}}$$
Подробное решение
Не могу найти шаги в поиске этой производной.
Но производная
Ответ:
График
Первая производная
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/ ___\
atan\\/ x / // 2 \ / ___\ / 7 \ \
7 |\-7 - 7*cot (x)/*atan\\/ x / log\cot (x)/ |
cot (x) *|---------------------------- + ---------------|
| cot(x) ___ |
\ 2*\/ x *(1 + x)/
$$\left(\frac{\operatorname{atan}{\left (\sqrt{x} \right )}}{\cot{\left (x \right )}} \left(- 7 \cot^{2}{\left (x \right )} - 7\right) + \frac{\log{\left (\cot^{7}{\left (x \right )} \right )}}{2 \sqrt{x} \left(x + 1\right)}\right) \left(\cot^{7}{\left (x \right )}\right)^{\operatorname{atan}{\left (\sqrt{x} \right )}}$$
Вторая производная
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/ 2 \
|/ / 7 \ / 2 \ / ___\\ |
|| log\cot (x)/ 14*\1 + cot (x)/*atan\\/ x /| |
/ ___\ ||- ------------- + ----------------------------| 2 |
atan\\/ x / || ___ cot(x) | / 2 \ / ___\ / 7 \ / 7 \ / 2 \ |
7 |\ \/ x *(1 + x) / / 2 \ / ___\ 7*\1 + cot (x)/ *atan\\/ x / log\cot (x)/ log\cot (x)/ 7*\1 + cot (x)/ |
cot (x) *|------------------------------------------------- + 14*\1 + cot (x)/*atan\\/ x / - ---------------------------- - ---------------- - -------------- - --------------------|
| 4 2 ___ 2 3/2 ___ |
\ cot (x) 2*\/ x *(1 + x) 4*x *(1 + x) \/ x *(1 + x)*cot(x)/
$$\left(\frac{1}{4} \left(\frac{14 \operatorname{atan}{\left (\sqrt{x} \right )}}{\cot{\left (x \right )}} \left(\cot^{2}{\left (x \right )} + 1\right) - \frac{\log{\left (\cot^{7}{\left (x \right )} \right )}}{\sqrt{x} \left(x + 1\right)}\right)^{2} - \frac{7 \left(\cot^{2}{\left (x \right )} + 1\right)^{2}}{\cot^{2}{\left (x \right )}} \operatorname{atan}{\left (\sqrt{x} \right )} + 14 \left(\cot^{2}{\left (x \right )} + 1\right) \operatorname{atan}{\left (\sqrt{x} \right )} - \frac{7 \cot^{2}{\left (x \right )} + 7}{\sqrt{x} \left(x + 1\right) \cot{\left (x \right )}} - \frac{\log{\left (\cot^{7}{\left (x \right )} \right )}}{2 \sqrt{x} \left(x + 1\right)^{2}} - \frac{\log{\left (\cot^{7}{\left (x \right )} \right )}}{4 x^{\frac{3}{2}} \left(x + 1\right)}\right) \left(\cot^{7}{\left (x \right )}\right)^{\operatorname{atan}{\left (\sqrt{x} \right )}}$$
Третья производная
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/ 3 / 2 \ \
| / / 7 \ / 2 \ / ___\\ / / 7 \ / 2 \ / ___\\ | / 7 \ / 7 \ / 2 \ / ___\ / 2 \ | |
| | log\cot (x)/ 14*\1 + cot (x)/*atan\\/ x /| | log\cot (x)/ 14*\1 + cot (x)/*atan\\/ x /| | / 2 \ / ___\ log\cot (x)/ 2*log\cot (x)/ 28*\1 + cot (x)/ *atan\\/ x / 28*\1 + cot (x)/ | |
/ ___\ | |- ------------- + ----------------------------| 3*|- ------------- + ----------------------------|*|- 56*\1 + cot (x)/*atan\\/ x / + ------------ + -------------- + ----------------------------- + --------------------| 3 2 2 |
atan\\/ x / | | ___ cot(x) | | ___ cot(x) | | 3/2 ___ 2 2 ___ | / 7 \ / 7 \ / 2 \ / ___\ / 2 \ / 2 \ / ___\ / 7 \ / 2 \ / 2 \ / 2 \ |
7 | \ \/ x *(1 + x) / \ \/ x *(1 + x) / \ x *(1 + x) \/ x *(1 + x) cot (x) \/ x *(1 + x)*cot(x)/ log\cot (x)/ log\cot (x)/ / 2 \ / ___\ 14*\1 + cot (x)/ *atan\\/ x / 21*\1 + cot (x)/ 28*\1 + cot (x)/ *atan\\/ x / 3*log\cot (x)/ 21*\1 + cot (x)/ 21*\1 + cot (x)/ 21*\1 + cot (x)/ |
cot (x) *|- ------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + -------------- + --------------- - 28*\1 + cot (x)/*atan\\/ x /*cot(x) - ----------------------------- + ---------------- + ----------------------------- + -------------- - ----------------------- + ----------------------- + ---------------------|
| 8 8 ___ 3 3/2 2 3 ___ cot(x) 5/2 ___ 2 ___ 2 3/2 |
\ \/ x *(1 + x) 2*x *(1 + x) cot (x) \/ x *(1 + x) 8*x *(1 + x) 2*\/ x *(1 + x)*cot (x) 2*\/ x *(1 + x) *cot(x) 4*x *(1 + x)*cot(x)/
$$\left(- \frac{1}{8} \left(\frac{14 \operatorname{atan}{\left (\sqrt{x} \right )}}{\cot{\left (x \right )}} \left(\cot^{2}{\left (x \right )} + 1\right) - \frac{\log{\left (\cot^{7}{\left (x \right )} \right )}}{\sqrt{x} \left(x + 1\right)}\right)^{3} + \frac{3}{8} \left(\frac{14 \operatorname{atan}{\left (\sqrt{x} \right )}}{\cot{\left (x \right )}} \left(\cot^{2}{\left (x \right )} + 1\right) - \frac{\log{\left (\cot^{7}{\left (x \right )} \right )}}{\sqrt{x} \left(x + 1\right)}\right) \left(\frac{28 \left(\cot^{2}{\left (x \right )} + 1\right)^{2}}{\cot^{2}{\left (x \right )}} \operatorname{atan}{\left (\sqrt{x} \right )} - 56 \left(\cot^{2}{\left (x \right )} + 1\right) \operatorname{atan}{\left (\sqrt{x} \right )} + \frac{28 \cot^{2}{\left (x \right )} + 28}{\sqrt{x} \left(x + 1\right) \cot{\left (x \right )}} + \frac{2 \log{\left (\cot^{7}{\left (x \right )} \right )}}{\sqrt{x} \left(x + 1\right)^{2}} + \frac{\log{\left (\cot^{7}{\left (x \right )} \right )}}{x^{\frac{3}{2}} \left(x + 1\right)}\right) - \frac{14 \left(\cot^{2}{\left (x \right )} + 1\right)^{3}}{\cot^{3}{\left (x \right )}} \operatorname{atan}{\left (\sqrt{x} \right )} + \frac{28 \left(\cot^{2}{\left (x \right )} + 1\right)^{2}}{\cot{\left (x \right )}} \operatorname{atan}{\left (\sqrt{x} \right )} - 28 \left(\cot^{2}{\left (x \right )} + 1\right) \cot{\left (x \right )} \operatorname{atan}{\left (\sqrt{x} \right )} - \frac{21 \left(\cot^{2}{\left (x \right )} + 1\right)^{2}}{2 \sqrt{x} \left(x + 1\right) \cot^{2}{\left (x \right )}} + \frac{21 \cot^{2}{\left (x \right )} + 21}{\sqrt{x} \left(x + 1\right)} + \frac{21 \cot^{2}{\left (x \right )} + 21}{2 \sqrt{x} \left(x + 1\right)^{2} \cot{\left (x \right )}} + \frac{\log{\left (\cot^{7}{\left (x \right )} \right )}}{\sqrt{x} \left(x + 1\right)^{3}} + \frac{21 \cot^{2}{\left (x \right )} + 21}{4 x^{\frac{3}{2}} \left(x + 1\right) \cot{\left (x \right )}} + \frac{\log{\left (\cot^{7}{\left (x \right )} \right )}}{2 x^{\frac{3}{2}} \left(x + 1\right)^{2}} + \frac{3 \log{\left (\cot^{7}{\left (x \right )} \right )}}{8 x^{\frac{5}{2}} \left(x + 1\right)}\right) \left(\cot^{7}{\left (x \right )}\right)^{\operatorname{atan}{\left (\sqrt{x} \right )}}$$