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The derivative of the function/ arcsin(sinx/(sqrt(1+sin^2x)))

Derivative of arcsin(sinx/(sqrt(1+sin^2x)))

Function f() - derivative -N order at the point
v

The graph:

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The solution

You have entered [src] 
    /     sin(x)     \
asin|----------------|
    |   _____________|
    |  /        2    |
    \\/  1 + sin (x) /
asin⁡(sin⁡(x)sin⁡2(x)+1)\operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{\sqrt{\sin^{2}{\left(x \right)} + 1}} \right)} 
asin(sin(x)/sqrt(1 + sin(x)^2))
The graph
02468-8-6-4-2-10102-2
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
                       2           
     cos(x)         sin (x)*cos(x) 
---------------- - ----------------
   _____________                3/2
  /        2       /       2   \   
\/  1 + sin (x)    \1 + sin (x)/   
-----------------------------------
            _________________      
           /          2            
          /        sin (x)         
         /   1 - -----------       
        /               2          
      \/         1 + sin (x)
cos⁡(x)sin⁡2(x)+1−sin⁡2(x)cos⁡(x)(sin⁡2(x)+1)321−sin⁡2(x)sin⁡2(x)+1\frac{\frac{\cos{\left(x \right)}}{\sqrt{\sin^{2}{\left(x \right)} + 1}} - \frac{\sin^{2}{\left(x \right)} \cos{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{\frac{3}{2}}}}{\sqrt{1 - \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1}}}
Simplify
The second derivative [src] 
/           2              2            2       2            _________________        \       
|        sin (x)      3*cos (x)    3*cos (x)*sin (x)        /          2              |       
|  1 - ----------- + ----------- - -----------------       /        sin (x)       2   |       
|             2             2                     2       /   1 - ----------- *cos (x)|       
|      1 + sin (x)   1 + sin (x)     /       2   \       /               2            |       
|                                    \1 + sin (x)/     \/         1 + sin (x)         |       
|- ------------------------------------------------- + -------------------------------|*sin(x)
|                     _________________                                 2             |       
|                    /          2                                1 + sin (x)          |       
|                   /        sin (x)                                                  |       
|                  /   1 - -----------                                                |       
|                 /               2                                                   |       
\               \/         1 + sin (x)                                                /       
----------------------------------------------------------------------------------------------
                                          _____________                                       
                                         /        2                                           
                                       \/  1 + sin (x)
(1−sin⁡2(x)sin⁡2(x)+1cos⁡2(x)sin⁡2(x)+1−1−sin⁡2(x)sin⁡2(x)+1+3cos⁡2(x)sin⁡2(x)+1−3sin⁡2(x)cos⁡2(x)(sin⁡2(x)+1)21−sin⁡2(x)sin⁡2(x)+1)sin⁡(x)sin⁡2(x)+1\frac{\left(\frac{\sqrt{1 - \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1}} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} - \frac{1 - \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} + \frac{3 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} - \frac{3 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}}}{\sqrt{1 - \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1}}}\right) \sin{\left(x \right)}}{\sqrt{\sin^{2}{\left(x \right)} + 1}}
Simplify
The third derivative [src] 
/             2            2              4               2       2            2       4                               4             2       2           2       4                /         2              2            2       2   \           _________________                \       
|       10*sin (x)    3*cos (x)      9*sin (x)      18*cos (x)*sin (x)   15*cos (x)*sin (x)      2         2        sin (x)     5*cos (x)*sin (x)   4*cos (x)*sin (x)        2    |      sin (x)      3*cos (x)    3*cos (x)*sin (x)|          /          2                      |       
|  1 - ----------- + ----------- + -------------- - ------------------ + ------------------   cos (x) - sin (x) + ----------- - ----------------- + -----------------   2*sin (x)*|1 - ----------- + ----------- - -----------------|         /        sin (x)       2       2   |       
|             2             2                   2                  2                    3                                2                2                        2              |           2             2                     2 |   3*   /   1 - ----------- *cos (x)*sin (x)|       
|      1 + sin (x)   1 + sin (x)   /       2   \      /       2   \        /       2   \                          1 + sin (x)      1 + sin (x)        /       2   \               |    1 + sin (x)   1 + sin (x)     /       2   \  |       /               2                    |       
|                                  \1 + sin (x)/      \1 + sin (x)/        \1 + sin (x)/                                                              \1 + sin (x)/               \                                  \1 + sin (x)/  /     \/         1 + sin (x)                 |       
|- ---------------------------------------------------------------------------------------- + ----------------------------------------------------------------------- - ------------------------------------------------------------- + -----------------------------------------|*cos(x)
|                                        _________________                                                                         _________________                                                    _________________                                          2             |       
|                                       /          2                                                                              /          2                                                         /          2                                   /       2   \              |       
|                                      /        sin (x)                                                        /       2   \     /        sin (x)                                   /       2   \     /        sin (x)                                \1 + sin (x)/              |       
|                                     /   1 - -----------                                                      \1 + sin (x)/*   /   1 - -----------                                 \1 + sin (x)/*   /   1 - -----------                                                         |       
|                                    /               2                                                                         /               2                                                    /               2                                                            |       
\                                  \/         1 + sin (x)                                                                    \/         1 + sin (x)                                               \/         1 + sin (x)                                                         /       
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                        _____________                                                                                                                                    
                                                                                                                                       /        2                                                                                                                                        
                                                                                                                                     \/  1 + sin (x)
(31−sin⁡2(x)sin⁡2(x)+1sin⁡2(x)cos⁡2(x)(sin⁡2(x)+1)2−1−10sin⁡2(x)sin⁡2(x)+1+3cos⁡2(x)sin⁡2(x)+1+9sin⁡4(x)(sin⁡2(x)+1)2−18sin⁡2(x)cos⁡2(x)(sin⁡2(x)+1)2+15sin⁡4(x)cos⁡2(x)(sin⁡2(x)+1)31−sin⁡2(x)sin⁡2(x)+1−2(1−sin⁡2(x)sin⁡2(x)+1+3cos⁡2(x)sin⁡2(x)+1−3sin⁡2(x)cos⁡2(x)(sin⁡2(x)+1)2)sin⁡2(x)1−sin⁡2(x)sin⁡2(x)+1(sin⁡2(x)+1)+−sin⁡2(x)+cos⁡2(x)+sin⁡4(x)sin⁡2(x)+1−5sin⁡2(x)cos⁡2(x)sin⁡2(x)+1+4sin⁡4(x)cos⁡2(x)(sin⁡2(x)+1)21−sin⁡2(x)sin⁡2(x)+1(sin⁡2(x)+1))cos⁡(x)sin⁡2(x)+1\frac{\left(\frac{3 \sqrt{1 - \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1}} \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}} - \frac{1 - \frac{10 \sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} + \frac{3 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} + \frac{9 \sin^{4}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}} - \frac{18 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}} + \frac{15 \sin^{4}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{3}}}{\sqrt{1 - \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1}}} - \frac{2 \left(1 - \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} + \frac{3 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} - \frac{3 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}}\right) \sin^{2}{\left(x \right)}}{\sqrt{1 - \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1}} \left(\sin^{2}{\left(x \right)} + 1\right)} + \frac{- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} + \frac{\sin^{4}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} - \frac{5 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} + \frac{4 \sin^{4}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}}}{\sqrt{1 - \frac{\sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1}} \left(\sin^{2}{\left(x \right)} + 1\right)}\right) \cos{\left(x \right)}}{\sqrt{\sin^{2}{\left(x \right)} + 1}}
Simplify
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