Производная (cos(x)^(1/5))*((atan(x)^4)*x)

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↑ Функция f () ? - производная -го порядка

Решение

Вы ввели
[LaTeX]
5 ________     4     
\/ cos(x) *atan (x)*x
$$x \operatorname{atan}^{4}{\left (x \right )} \sqrt[5]{\cos{\left (x \right )}}$$
График
[LaTeX]
Первая производная
[LaTeX]
           /                   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
[LaTeX]
         /             /                        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
[LaTeX]
/                                       /                                        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 )}$$