Производная atan(sin(x)+1)

Вы ввели

atan(sin(x) + 1)

$$\operatorname{atan}{\left (\sin{\left (x \right )} + 1 \right )}$$

Первая производная

      cos(x)     
-----------------
                2
1 + (sin(x) + 1) 

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

Вторая производная

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

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

Третья производная

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

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