Производная atan(sin(x))

Вы ввели

atan(sin(x))

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

d               
--(atan(sin(x)))
dx              

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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

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