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arcsin((sinx)/(1+sin^2x))
Derivada de la función/ arcsin/ sin^2/ (sinx)/ arcsin((sinx)/(1+sin^2x))

Derivada de arcsin((sinx)/(1+sin^2x))

Función f() - derivada -er orden en el punto
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
    /   sin(x)  \
asin|-----------|
    |       2   |
    \1 + sin (x)/
asin⁡(sin⁡(x)sin⁡2(x)+1)\operatorname{asin}{\left(\frac{\sin{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} \right)} 
asin(sin(x)/(1 + sin(x)^2))
Gráfica
02468-8-6-4-2-10102-2
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
                   2          
   cos(x)     2*sin (x)*cos(x)
----------- - ----------------
       2                    2 
1 + sin (x)    /       2   \  
               \1 + sin (x)/  
------------------------------
        ____________________  
       /           2          
      /         sin (x)       
     /   1 - --------------   
    /                     2   
   /         /       2   \    
 \/          \1 + sin (x)/
cos⁡(x)sin⁡2(x)+1−2sin⁡2(x)cos⁡(x)(sin⁡2(x)+1)21−sin⁡2(x)(sin⁡2(x)+1)2\frac{\frac{\cos{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} - \frac{2 \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}}}{\sqrt{1 - \frac{\sin^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}}}}
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Segunda derivada [src] 
/                                                                           2            \       
|                                                          /          2    \             |       
|                                                          |     2*sin (x) |     2       |       
|                                                          |1 - -----------| *cos (x)    |       
|           2             2            2       2           |           2   |             |       
|      6*cos (x)     2*sin (x)    8*cos (x)*sin (x)        \    1 + sin (x)/             |       
|-1 - ----------- + ----------- + ----------------- + -----------------------------------|*sin(x)
|            2             2                     2                 2 /          2       \|       
|     1 + sin (x)   1 + sin (x)     /       2   \     /       2   \  |       sin (x)    ||       
|                                   \1 + sin (x)/     \1 + sin (x)/ *|1 - --------------||       
|                                                                    |                 2||       
|                                                                    |    /       2   \ ||       
\                                                                    \    \1 + sin (x)/ //       
-------------------------------------------------------------------------------------------------
                                                 ____________________                            
                                                /           2                                    
                            /       2   \      /         sin (x)                                 
                            \1 + sin (x)/*    /   1 - --------------                             
                                             /                     2                             
                                            /         /       2   \                              
                                          \/          \1 + sin (x)/
(−1+2sin⁡2(x)sin⁡2(x)+1−6cos⁡2(x)sin⁡2(x)+1+8sin⁡2(x)cos⁡2(x)(sin⁡2(x)+1)2+(1−2sin⁡2(x)sin⁡2(x)+1)2cos⁡2(x)(1−sin⁡2(x)(sin⁡2(x)+1)2)(sin⁡2(x)+1)2)sin⁡(x)1−sin⁡2(x)(sin⁡2(x)+1)2(sin⁡2(x)+1)\frac{\left(-1 + \frac{2 \sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} - \frac{6 \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{\left(1 - \frac{2 \sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1}\right)^{2} \cos^{2}{\left(x \right)}}{\left(1 - \frac{\sin^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}}\right) \left(\sin^{2}{\left(x \right)} + 1\right)^{2}}\right) \sin{\left(x \right)}}{\sqrt{1 - \frac{\sin^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}}} \left(\sin^{2}{\left(x \right)} + 1\right)}
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Tercera derivada [src] 
/                                                                                            /          2    \ /                          4             2       2            2       4   \             /          2    \ /          2             2            2       2   \                      3                \       
|                                                                                            |     2*sin (x) | |   2         2       2*sin (x)    10*cos (x)*sin (x)   12*cos (x)*sin (x)|        2    |     2*sin (x) | |     2*sin (x)     6*cos (x)    8*cos (x)*sin (x)|     /          2    \                 |       
|                                                                                            |1 - -----------|*|cos (x) - sin (x) + ----------- - ------------------ + ------------------|   2*sin (x)*|1 - -----------|*|1 - ----------- + ----------- - -----------------|     |     2*sin (x) |     2       2   |       
|                                                                                            |           2   | |                           2                2                         2  |             |           2   | |           2             2                     2 |   3*|1 - -----------| *cos (x)*sin (x)|       
|             4              2              2            2       4            2       2      \    1 + sin (x)/ |                    1 + sin (x)      1 + sin (x)         /       2   \   |             \    1 + sin (x)/ |    1 + sin (x)   1 + sin (x)     /       2   \  |     |           2   |                 |       
|       24*sin (x)      6*cos (x)     20*sin (x)   48*cos (x)*sin (x)   48*cos (x)*sin (x)                     \                                                         \1 + sin (x)/   /                               \                                  \1 + sin (x)/  /     \    1 + sin (x)/                 |       
|-1 - -------------- - ----------- + ----------- - ------------------ + ------------------ + --------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------- + ------------------------------------|*cos(x)
|                  2          2             2                     3                    2                                               2 /          2       \                                                                   2 /          2       \                                                            2|       
|     /       2   \    1 + sin (x)   1 + sin (x)     /       2   \        /       2   \                                   /       2   \  |       sin (x)    |                                                      /       2   \  |       sin (x)    |                                      4 /          2       \ |       
|     \1 + sin (x)/                                  \1 + sin (x)/        \1 + sin (x)/                                   \1 + sin (x)/ *|1 - --------------|                                                      \1 + sin (x)/ *|1 - --------------|                         /       2   \  |       sin (x)    | |       
|                                                                                                                                        |                 2|                                                                     |                 2|                         \1 + sin (x)/ *|1 - --------------| |       
|                                                                                                                                        |    /       2   \ |                                                                     |    /       2   \ |                                        |                 2| |       
|                                                                                                                                        \    \1 + sin (x)/ /                                                                     \    \1 + sin (x)/ /                                        |    /       2   \ | |       
\                                                                                                                                                                                                                                                                                             \    \1 + sin (x)/ / /       
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                              ____________________                                                                                                                                         
                                                                                                                                                             /           2                                                                                                                                                 
                                                                                                                                         /       2   \      /         sin (x)                                                                                                                                              
                                                                                                                                         \1 + sin (x)/*    /   1 - --------------                                                                                                                                          
                                                                                                                                                          /                     2                                                                                                                                          
                                                                                                                                                         /         /       2   \                                                                                                                                           
                                                                                                                                                       \/          \1 + sin (x)/
(−1+20sin⁡2(x)sin⁡2(x)+1−6cos⁡2(x)sin⁡2(x)+1−24sin⁡4(x)(sin⁡2(x)+1)2+48sin⁡2(x)cos⁡2(x)(sin⁡2(x)+1)2−48sin⁡4(x)cos⁡2(x)(sin⁡2(x)+1)3−2(1−2sin⁡2(x)sin⁡2(x)+1)(1−2sin⁡2(x)sin⁡2(x)+1+6cos⁡2(x)sin⁡2(x)+1−8sin⁡2(x)cos⁡2(x)(sin⁡2(x)+1)2)sin⁡2(x)(1−sin⁡2(x)(sin⁡2(x)+1)2)(sin⁡2(x)+1)2+(1−2sin⁡2(x)sin⁡2(x)+1)(−sin⁡2(x)+cos⁡2(x)+2sin⁡4(x)sin⁡2(x)+1−10sin⁡2(x)cos⁡2(x)sin⁡2(x)+1+12sin⁡4(x)cos⁡2(x)(sin⁡2(x)+1)2)(1−sin⁡2(x)(sin⁡2(x)+1)2)(sin⁡2(x)+1)2+3(1−2sin⁡2(x)sin⁡2(x)+1)3sin⁡2(x)cos⁡2(x)(1−sin⁡2(x)(sin⁡2(x)+1)2)2(sin⁡2(x)+1)4)cos⁡(x)1−sin⁡2(x)(sin⁡2(x)+1)2(sin⁡2(x)+1)\frac{\left(-1 + \frac{20 \sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} - \frac{6 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} - \frac{24 \sin^{4}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}} + \frac{48 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}} - \frac{48 \sin^{4}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{3}} - \frac{2 \left(1 - \frac{2 \sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1}\right) \left(1 - \frac{2 \sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} + \frac{6 \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) \sin^{2}{\left(x \right)}}{\left(1 - \frac{\sin^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}}\right) \left(\sin^{2}{\left(x \right)} + 1\right)^{2}} + \frac{\left(1 - \frac{2 \sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1}\right) \left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} + \frac{2 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} - \frac{10 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} + \frac{12 \sin^{4}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}}\right)}{\left(1 - \frac{\sin^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}}\right) \left(\sin^{2}{\left(x \right)} + 1\right)^{2}} + \frac{3 \left(1 - \frac{2 \sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1}\right)^{3} \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(1 - \frac{\sin^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}}\right)^{2} \left(\sin^{2}{\left(x \right)} + 1\right)^{4}}\right) \cos{\left(x \right)}}{\sqrt{1 - \frac{\sin^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}}} \left(\sin^{2}{\left(x \right)} + 1\right)}
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Gráfico
Derivada de arcsin((sinx)/(1+sin^2x))
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