Производная log(atan(sin(x)))

Вы ввели

log(atan(sin(x)))

$$\log{\left (\operatorname{atan}{\left (\sin{\left (x \right )} \right )} \right )}$$

Первая производная

          cos(x)          
--------------------------
/       2   \             
\1 + sin (x)/*atan(sin(x))

$$\frac{\cos{\left (x \right )}}{\left(\sin^{2}{\left (x \right )} + 1\right) \operatorname{atan}{\left (\sin{\left (x \right )} \right )}}$$

Вторая производная

 /            2                     2                   \ 
 |         cos (x)             2*cos (x)*sin(x)         | 
-|-------------------------- + ---------------- + sin(x)| 
 |/       2   \                         2               | 
 \\1 + sin (x)/*atan(sin(x))     1 + sin (x)            / 
----------------------------------------------------------
                /       2   \                             
                \1 + sin (x)/*atan(sin(x))                

$$- \frac{1}{\left(\sin^{2}{\left (x \right )} + 1\right) \operatorname{atan}{\left (\sin{\left (x \right )} \right )}} \left(\sin{\left (x \right )} + \frac{2 \sin{\left (x \right )} \cos^{2}{\left (x \right )}}{\sin^{2}{\left (x \right )} + 1} + \frac{\cos^{2}{\left (x \right )}}{\left(\sin^{2}{\left (x \right )} + 1\right) \operatorname{atan}{\left (\sin{\left (x \right )} \right )}}\right)$$

Третья производная

/           2             2                     2                                                  2       2                 2               \       
|      2*cos (x)     6*sin (x)             2*cos (x)                      3*sin(x)            8*cos (x)*sin (x)         6*cos (x)*sin(x)     |       
|-1 - ----------- + ----------- + ---------------------------- + -------------------------- + ----------------- + ---------------------------|*cos(x)
|            2             2                   2                 /       2   \                               2                 2             |       
|     1 + sin (x)   1 + sin (x)   /       2   \      2           \1 + sin (x)/*atan(sin(x))     /       2   \     /       2   \              |       
\                                 \1 + sin (x)/ *atan (sin(x))                                  \1 + sin (x)/     \1 + sin (x)/ *atan(sin(x))/       
-----------------------------------------------------------------------------------------------------------------------------------------------------
                                                              /       2   \                                                                          
                                                              \1 + sin (x)/*atan(sin(x))                                                             

$$\frac{\cos{\left (x \right )}}{\left(\sin^{2}{\left (x \right )} + 1\right) \operatorname{atan}{\left (\sin{\left (x \right )} \right )}} \left(-1 + \frac{6 \sin^{2}{\left (x \right )}}{\sin^{2}{\left (x \right )} + 1} + \frac{3 \sin{\left (x \right )}}{\left(\sin^{2}{\left (x \right )} + 1\right) \operatorname{atan}{\left (\sin{\left (x \right )} \right )}} - \frac{2 \cos^{2}{\left (x \right )}}{\sin^{2}{\left (x \right )} + 1} + \frac{8 \sin^{2}{\left (x \right )} \cos^{2}{\left (x \right )}}{\left(\sin^{2}{\left (x \right )} + 1\right)^{2}} + \frac{6 \sin{\left (x \right )} \cos^{2}{\left (x \right )}}{\left(\sin^{2}{\left (x \right )} + 1\right)^{2} \operatorname{atan}{\left (\sin{\left (x \right )} \right )}} + \frac{2 \cos^{2}{\left (x \right )}}{\left(\sin^{2}{\left (x \right )} + 1\right)^{2} \operatorname{atan}^{2}{\left (\sin{\left (x \right )} \right )}}\right)$$

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