Производная (cos(x)^(1/5))*((atan(x)^4)*x)
Решение
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5 ________ 4 \/ cos(x) *atan (x)*x
$$x \operatorname{atan}^{4}{\left (x \right )} \sqrt[5]{\cos{\left (x \right )}}$$
График
Первая производная
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/ 3 \ 4
5 ________ | 4 4*x*atan (x)| x*atan (x)*sin(x)
\/ cos(x) *|atan (x) + ------------| - -----------------
| 2 | 4/5
\ 1 + x / 5*cos (x)
$$- \frac{x \sin{\left (x \right )} \operatorname{atan}^{4}{\left (x \right )}}{5 \cos^{\frac{4}{5}}{\left (x \right )}} + \left(\frac{4 x \operatorname{atan}^{3}{\left (x \right )}}{x^{2} + 1} + \operatorname{atan}^{4}{\left (x \right )}\right) \sqrt[5]{\cos{\left (x \right )}}$$
Вторая производная
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/ / 2 \ \
| 5 ________ | 3*x 2*x *atan(x)| / 4*x \ |
|4*\/ cos(x) *|2*atan(x) + ------ - ------------| |------ + atan(x)|*atan(x)*sin(x) |
| | 2 2 | 2 5 ________ 2 2 2 | 2 | |
2 | \ 1 + x 1 + x / x*atan (x)*\/ cos(x) atan (x)*sin(x) 4*x*atan (x)*sin (x) \1 + x / 4*x*atan(x)*sin(x) |
atan (x)*|------------------------------------------------ - --------------------- - --------------- - -------------------- - --------------------------------- - --------------------|
| 2 5 4/5 9/5 4/5 / 2\ 4/5 |
\ 1 + x 5*cos (x) 25*cos (x) 5*cos (x) 5*\1 + x /*cos (x)/
$$\left(- \frac{4 x \sin^{2}{\left (x \right )} \operatorname{atan}^{2}{\left (x \right )}}{25 \cos^{\frac{9}{5}}{\left (x \right )}} - \frac{x}{5} \sqrt[5]{\cos{\left (x \right )}} \operatorname{atan}^{2}{\left (x \right )} - \frac{4 x \sin{\left (x \right )} \operatorname{atan}{\left (x \right )}}{5 \left(x^{2} + 1\right) \cos^{\frac{4}{5}}{\left (x \right )}} - \frac{\sin{\left (x \right )} \operatorname{atan}{\left (x \right )}}{5 \cos^{\frac{4}{5}}{\left (x \right )}} \left(\frac{4 x}{x^{2} + 1} + \operatorname{atan}{\left (x \right )}\right) - \frac{\sin{\left (x \right )} \operatorname{atan}^{2}{\left (x \right )}}{5 \cos^{\frac{4}{5}}{\left (x \right )}} + \frac{4 \sqrt[5]{\cos{\left (x \right )}}}{x^{2} + 1} \left(- \frac{2 x^{2} \operatorname{atan}{\left (x \right )}}{x^{2} + 1} + \frac{3 x}{x^{2} + 1} + 2 \operatorname{atan}{\left (x \right )}\right)\right) \operatorname{atan}^{2}{\left (x \right )}$$
Третья производная
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/ / 2 3 2 \ / 2 \ \ | 5 ________ | 2 6*x 18*x *atan(x) 8*x *atan (x)| 2 5 ________ / 4*x \ 2 2 / 4*x \ | 3*x 2*x *atan(x)| | | 4*\/ cos(x) *|9*atan(x) - 8*x*atan (x) + ------ - ------------- + -------------| atan (x)*\/ cos(x) *|------ + atan(x)| 4*atan (x)*sin (x)*|------ + atan(x)| 8*|2*atan(x) + ------ - ------------|*atan(x)*sin(x) | | 3 5 ________ | 2 2 2 | 3 2 | 2 | 3 3 2 5 ________ 2 3 | 2 | 2 2 | 2 2 | 2 2 | | 2*atan (x)*\/ cos(x) \ 1 + x 1 + x 1 + x / 8*atan (x)*sin (x) \1 + x / 36*x*atan (x)*sin (x) 8*x*atan (x)*\/ cos(x) 8*atan (x)*sin(x) 7*x*atan (x)*sin(x) \1 + x / 32*x*atan (x)*sin (x) 12*x*atan(x)*sin(x) \ 1 + x 1 + x / 8*x *atan (x)*sin(x)| |- --------------------- + -------------------------------------------------------------------------------- - ------------------ - -------------------------------------- - --------------------- - ----------------------- - -------------------- - ------------------- - ------------------------------------- - --------------------- - --------------------- - ---------------------------------------------------- + ---------------------|*atan(x) | 5 2 9/5 5 14/5 / 2\ / 2\ 4/5 4/5 9/5 / 2\ 9/5 2 / 2\ 4/5 2 | | / 2\ 25*cos (x) 125*cos (x) 5*\1 + x / 5*\1 + x /*cos (x) 25*cos (x) 25*cos (x) 25*\1 + x /*cos (x) / 2\ 4/5 5*\1 + x /*cos (x) / 2\ 4/5 | \ \1 + x / 5*\1 + x / *cos (x) 5*\1 + x / *cos (x)/
$$\left(\frac{8 x^{2} \sin{\left (x \right )} \operatorname{atan}^{2}{\left (x \right )}}{5 \left(x^{2} + 1\right)^{2} \cos^{\frac{4}{5}}{\left (x \right )}} - \frac{36 x \sin^{3}{\left (x \right )} \operatorname{atan}^{3}{\left (x \right )}}{125 \cos^{\frac{14}{5}}{\left (x \right )}} - \frac{7 x \sin{\left (x \right )} \operatorname{atan}^{3}{\left (x \right )}}{25 \cos^{\frac{4}{5}}{\left (x \right )}} - \frac{32 x \sin^{2}{\left (x \right )} \operatorname{atan}^{2}{\left (x \right )}}{25 \left(x^{2} + 1\right) \cos^{\frac{9}{5}}{\left (x \right )}} - \frac{8 x \operatorname{atan}^{2}{\left (x \right )}}{5 x^{2} + 5} \sqrt[5]{\cos{\left (x \right )}} - \frac{12 x \sin{\left (x \right )} \operatorname{atan}{\left (x \right )}}{5 \left(x^{2} + 1\right)^{2} \cos^{\frac{4}{5}}{\left (x \right )}} - \frac{4 \sin^{2}{\left (x \right )} \operatorname{atan}^{2}{\left (x \right )}}{25 \cos^{\frac{9}{5}}{\left (x \right )}} \left(\frac{4 x}{x^{2} + 1} + \operatorname{atan}{\left (x \right )}\right) - \frac{1}{5} \left(\frac{4 x}{x^{2} + 1} + \operatorname{atan}{\left (x \right )}\right) \sqrt[5]{\cos{\left (x \right )}} \operatorname{atan}^{2}{\left (x \right )} - \frac{8 \sin^{2}{\left (x \right )} \operatorname{atan}^{3}{\left (x \right )}}{25 \cos^{\frac{9}{5}}{\left (x \right )}} - \frac{2}{5} \sqrt[5]{\cos{\left (x \right )}} \operatorname{atan}^{3}{\left (x \right )} - \frac{8 \sin{\left (x \right )} \operatorname{atan}{\left (x \right )}}{5 \left(x^{2} + 1\right) \cos^{\frac{4}{5}}{\left (x \right )}} \left(- \frac{2 x^{2} \operatorname{atan}{\left (x \right )}}{x^{2} + 1} + \frac{3 x}{x^{2} + 1} + 2 \operatorname{atan}{\left (x \right )}\right) - \frac{8 \sin{\left (x \right )} \operatorname{atan}^{2}{\left (x \right )}}{5 \left(x^{2} + 1\right) \cos^{\frac{4}{5}}{\left (x \right )}} + \frac{4 \sqrt[5]{\cos{\left (x \right )}}}{\left(x^{2} + 1\right)^{2}} \left(\frac{8 x^{3} \operatorname{atan}^{2}{\left (x \right )}}{x^{2} + 1} - \frac{18 x^{2} \operatorname{atan}{\left (x \right )}}{x^{2} + 1} - 8 x \operatorname{atan}^{2}{\left (x \right )} + \frac{6 x}{x^{2} + 1} + 9 \operatorname{atan}{\left (x \right )}\right)\right) \operatorname{atan}{\left (x \right )}$$