Производная log(asin(x)/atan(x))

Вы ввели

   /asin(x)\
log|-------|
   \atan(x)/

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

Первая производная

/         1                 asin(x)     \        
|------------------- - -----------------|*atan(x)
|   ________           /     2\     2   |        
|  /      2            \1 + x /*atan (x)|        
\\/  1 - x  *atan(x)                    /        
-------------------------------------------------
                     asin(x)                     

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

Вторая производная

                     1            asin(x)               1            asin(x)                                                                             
              - ----------- + ----------------   - ----------- + ----------------                                                                        
                   ________   /     2\                ________   /     2\                                                                                
                  /      2    \1 + x /*atan(x)       /      2    \1 + x /*atan(x)                                                                        
     x          \/  1 - x                          \/  1 - x                                     2                     2*asin(x)           2*x*asin(x)   
----------- + -------------------------------- - -------------------------------- - ---------------------------- + ------------------ + -----------------
        3/2            ________                          /     2\                               ________                   2                    2        
/     2\              /      2                           \1 + x /*atan(x)           /     2\   /      2            /     2\      2      /     2\         
\1 - x /            \/  1 - x  *asin(x)                                             \1 + x /*\/  1 - x  *atan(x)   \1 + x / *atan (x)   \1 + x / *atan(x)
---------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                         asin(x)                                                                         

$$\frac{1}{\operatorname{asin}{\left (x \right )}} \left(\frac{2 x \operatorname{asin}{\left (x \right )}}{\left(x^{2} + 1\right)^{2} \operatorname{atan}{\left (x \right )}} + \frac{x}{\left(- x^{2} + 1\right)^{\frac{3}{2}}} - \frac{1}{\left(x^{2} + 1\right) \operatorname{atan}{\left (x \right )}} \left(\frac{\operatorname{asin}{\left (x \right )}}{\left(x^{2} + 1\right) \operatorname{atan}{\left (x \right )}} - \frac{1}{\sqrt{- x^{2} + 1}}\right) + \frac{2 \operatorname{asin}{\left (x \right )}}{\left(x^{2} + 1\right)^{2} \operatorname{atan}^{2}{\left (x \right )}} + \frac{\frac{\operatorname{asin}{\left (x \right )}}{\left(x^{2} + 1\right) \operatorname{atan}{\left (x \right )}} - \frac{1}{\sqrt{- x^{2} + 1}}}{\sqrt{- x^{2} + 1} \operatorname{asin}{\left (x \right )}} - \frac{2}{\sqrt{- x^{2} + 1} \left(x^{2} + 1\right) \operatorname{atan}{\left (x \right )}}\right)$$

Третья производная

                                                   /     x                     2                     2*asin(x)           2*x*asin(x)   \     /     x                     2                     2*asin(x)           2*x*asin(x)   \                         /       1            asin(x)     \                                      /       1            asin(x)     \                                                                               /       1            asin(x)     \                                     /       1            asin(x)     \
                                                 2*|----------- - ---------------------------- + ------------------ + -----------------|   2*|----------- - ---------------------------- + ------------------ + -----------------|                       2*|- ----------- + ----------------|                                    x*|- ----------- + ----------------|                                                                           2*x*|- ----------- + ----------------|                                   2*|- ----------- + ----------------|
                                                   |        3/2               ________                   2                    2        |     |        3/2               ________                   2                    2        |                         |     ________   /     2\        |                                      |     ________   /     2\        |                                                                               |     ________   /     2\        |                                     |     ________   /     2\        |
                     2                             |/     2\      /     2\   /      2            /     2\      2      /     2\         |     |/     2\      /     2\   /      2            /     2\      2      /     2\         |                         |    /      2    \1 + x /*atan(x)|                                      |    /      2    \1 + x /*atan(x)|                              2                                                |    /      2    \1 + x /*atan(x)|                                     |    /      2    \1 + x /*atan(x)|
     1            3*x           6*asin(x)          \\1 - x /      \1 + x /*\/  1 - x  *atan(x)   \1 + x / *atan (x)   \1 + x / *atan(x)/     \\1 - x /      \1 + x /*\/  1 - x  *atan(x)   \1 + x / *atan (x)   \1 + x / *atan(x)/       2*asin(x)         \  \/  1 - x                     /                 6                    \  \/  1 - x                     /      12*x*asin(x)         8*x *asin(x)                 3*x                    \  \/  1 - x                     /                6*x                  \  \/  1 - x                     /
----------- + ----------- - ------------------ - --------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------- + ----------------- + ------------------------------------ + ------------------------------ + ------------------------------------ - ------------------ - ----------------- - ---------------------------- + -------------------------------------- + ----------------------------- + ------------------------------------
        3/2           5/2           3                                                 ________                                                                                 /     2\                                                      2                    /      2\     2                       2    ________                            3/2                            3                    3                            3/2                             2                              2    ________                       ________                
/     2\      /     2\      /     2\      3                                          /      2                                                                                  \1 + x /*atan(x)                                      /     2\                     \-1 + x /*asin (x)            /     2\    /      2      2              /     2\                       /     2\      2      /     2\            /     2\ /     2\                        /     2\                       /     2\    /      2            /     2\   /      2                 
\1 - x /      \1 - x /      \1 + x / *atan (x)                                     \/  1 - x  *asin(x)                                                                                                                               \1 + x / *atan(x)                                          \1 + x / *\/  1 - x  *atan (x)           \1 - x /   *asin(x)            \1 + x / *atan (x)   \1 + x / *atan(x)   \1 + x /*\1 - x /   *atan(x)             \1 + x / *atan(x)              \1 + x / *\/  1 - x  *atan(x)   \1 + x /*\/  1 - x  *asin(x)*atan(x)
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                                                           asin(x)                                                                                                                                                                                                                                                                           

$$\frac{1}{\operatorname{asin}{\left (x \right )}} \left(- \frac{8 x^{2} \operatorname{asin}{\left (x \right )}}{\left(x^{2} + 1\right)^{3} \operatorname{atan}{\left (x \right )}} + \frac{3 x^{2}}{\left(- x^{2} + 1\right)^{\frac{5}{2}}} + \frac{2 x}{\left(x^{2} + 1\right)^{2} \operatorname{atan}{\left (x \right )}} \left(\frac{\operatorname{asin}{\left (x \right )}}{\left(x^{2} + 1\right) \operatorname{atan}{\left (x \right )}} - \frac{1}{\sqrt{- x^{2} + 1}}\right) - \frac{12 x \operatorname{asin}{\left (x \right )}}{\left(x^{2} + 1\right)^{3} \operatorname{atan}^{2}{\left (x \right )}} + \frac{6 x}{\sqrt{- x^{2} + 1} \left(x^{2} + 1\right)^{2} \operatorname{atan}{\left (x \right )}} + \frac{x \left(\frac{\operatorname{asin}{\left (x \right )}}{\left(x^{2} + 1\right) \operatorname{atan}{\left (x \right )}} - \frac{1}{\sqrt{- x^{2} + 1}}\right)}{\left(- x^{2} + 1\right)^{\frac{3}{2}} \operatorname{asin}{\left (x \right )}} - \frac{3 x}{\left(- x^{2} + 1\right)^{\frac{3}{2}} \left(x^{2} + 1\right) \operatorname{atan}{\left (x \right )}} + \frac{1}{\left(x^{2} + 1\right) \operatorname{atan}{\left (x \right )}} \left(\frac{4 x \operatorname{asin}{\left (x \right )}}{\left(x^{2} + 1\right)^{2} \operatorname{atan}{\left (x \right )}} + \frac{2 x}{\left(- x^{2} + 1\right)^{\frac{3}{2}}} + \frac{4 \operatorname{asin}{\left (x \right )}}{\left(x^{2} + 1\right)^{2} \operatorname{atan}^{2}{\left (x \right )}} - \frac{4}{\sqrt{- x^{2} + 1} \left(x^{2} + 1\right) \operatorname{atan}{\left (x \right )}}\right) + \frac{2 \operatorname{asin}{\left (x \right )}}{\left(x^{2} + 1\right)^{2} \operatorname{atan}{\left (x \right )}} - \frac{6 \operatorname{asin}{\left (x \right )}}{\left(x^{2} + 1\right)^{3} \operatorname{atan}^{3}{\left (x \right )}} + \frac{1}{\left(x^{2} - 1\right) \operatorname{asin}^{2}{\left (x \right )}} \left(\frac{2 \operatorname{asin}{\left (x \right )}}{\left(x^{2} + 1\right) \operatorname{atan}{\left (x \right )}} - \frac{2}{\sqrt{- x^{2} + 1}}\right) - \frac{1}{\sqrt{- x^{2} + 1} \operatorname{asin}{\left (x \right )}} \left(\frac{4 x \operatorname{asin}{\left (x \right )}}{\left(x^{2} + 1\right)^{2} \operatorname{atan}{\left (x \right )}} + \frac{2 x}{\left(- x^{2} + 1\right)^{\frac{3}{2}}} + \frac{4 \operatorname{asin}{\left (x \right )}}{\left(x^{2} + 1\right)^{2} \operatorname{atan}^{2}{\left (x \right )}} - \frac{4}{\sqrt{- x^{2} + 1} \left(x^{2} + 1\right) \operatorname{atan}{\left (x \right )}}\right) + \frac{\frac{2 \operatorname{asin}{\left (x \right )}}{\left(x^{2} + 1\right) \operatorname{atan}{\left (x \right )}} - \frac{2}{\sqrt{- x^{2} + 1}}}{\sqrt{- x^{2} + 1} \left(x^{2} + 1\right) \operatorname{asin}{\left (x \right )} \operatorname{atan}{\left (x \right )}} + \frac{6}{\sqrt{- x^{2} + 1} \left(x^{2} + 1\right)^{2} \operatorname{atan}^{2}{\left (x \right )}} + \frac{1}{\left(- x^{2} + 1\right)^{\frac{3}{2}}}\right)$$