Mister Exam

Derivative of y=ctg((√x)³)×arcsin²x

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

The graph:

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Piecewise:

The solution

You have entered [src]
   /     3\         
   |  ___ |     2   
cot\\/ x  /*asin (x)
$$\cot{\left(\left(\sqrt{x}\right)^{3} \right)} \operatorname{asin}^{2}{\left(x \right)}$$
cot((sqrt(x))^3)*asin(x)^2
The graph
The first derivative [src]
             /     3\                    /         /     3\\
             |  ___ |       ___     2    |        2|  ___ ||
2*asin(x)*cot\\/ x  /   3*\/ x *asin (x)*\-1 - cot \\/ x  //
--------------------- + ------------------------------------
        ________                         2                  
       /      2                                             
     \/  1 - x                                              
$$\frac{3 \sqrt{x} \left(- \cot^{2}{\left(\left(\sqrt{x}\right)^{3} \right)} - 1\right) \operatorname{asin}^{2}{\left(x \right)}}{2} + \frac{2 \cot{\left(\left(\sqrt{x}\right)^{3} \right)} \operatorname{asin}{\left(x \right)}}{\sqrt{1 - x^{2}}}$$
The second derivative [src]
                                                     /        /     3\\ /                 /     3\\                                     
                                                2    |       2|  ___ || |    1            |  ___ ||                                     
                                          3*asin (x)*\1 + cot \\/ x  //*|- ----- + 6*x*cot\\/ x  /|           /        /     3\\        
                               /     3\                                 |    ___                  |       ___ |       2|  ___ ||        
  /     1       x*asin(x) \    |  ___ |                                 \  \/ x                   /   6*\/ x *\1 + cot \\/ x  //*asin(x)
2*|- ------- + -----------|*cot\\/ x  / + --------------------------------------------------------- - ----------------------------------
  |        2           3/2|                                           4                                             ________            
  |  -1 + x    /     2\   |                                                                                        /      2             
  \            \1 - x /   /                                                                                      \/  1 - x              
$$- \frac{6 \sqrt{x} \left(\cot^{2}{\left(\left(\sqrt{x}\right)^{3} \right)} + 1\right) \operatorname{asin}{\left(x \right)}}{\sqrt{1 - x^{2}}} + \frac{3 \left(6 x \cot{\left(\left(\sqrt{x}\right)^{3} \right)} - \frac{1}{\sqrt{x}}\right) \left(\cot^{2}{\left(\left(\sqrt{x}\right)^{3} \right)} + 1\right) \operatorname{asin}^{2}{\left(x \right)}}{4} + 2 \left(\frac{x \operatorname{asin}{\left(x \right)}}{\left(1 - x^{2}\right)^{\frac{3}{2}}} - \frac{1}{x^{2} - 1}\right) \cot{\left(\left(\sqrt{x}\right)^{3} \right)}$$
The third derivative [src]
                                                                                                                            /        /     3\\ /               /     3\           /        /     3\\               /     3\\     /        /     3\\ /                 /     3\\        
                                                                                                                       2    |       2|  ___ || |   1           |  ___ |       3/2 |       2|  ___ ||       3/2    2|  ___ ||     |       2|  ___ || |    1            |  ___ ||        
                                                                                                                 3*asin (x)*\1 + cot \\/ x  //*|- ---- - 18*cot\\/ x  / + 18*x   *\1 + cot \\/ x  // + 36*x   *cot \\/ x  /|   9*\1 + cot \\/ x  //*|- ----- + 6*x*cot\\/ x  /|*asin(x)
  /                              2        \    /     3\           /        /     3\\                                                           |   3/2                                                                     |                        |    ___                  |        
  |  asin(x)        3*x       3*x *asin(x)|    |  ___ |       ___ |       2|  ___ || /     1       x*asin(x) \                                 \  x                                                                        /                        \  \/ x                   /        
2*|----------- + ---------- + ------------|*cot\\/ x  / - 9*\/ x *\1 + cot \\/ x  //*|- ------- + -----------| - ----------------------------------------------------------------------------------------------------------- + --------------------------------------------------------
  |        3/2            2           5/2 |                                          |        2           3/2|                                                        8                                                                                  ________                      
  |/     2\      /      2\    /     2\    |                                          |  -1 + x    /     2\   |                                                                                                                                          /      2                       
  \\1 - x /      \-1 + x /    \1 - x /    /                                          \            \1 - x /   /                                                                                                                                      2*\/  1 - x                        
$$- 9 \sqrt{x} \left(\frac{x \operatorname{asin}{\left(x \right)}}{\left(1 - x^{2}\right)^{\frac{3}{2}}} - \frac{1}{x^{2} - 1}\right) \left(\cot^{2}{\left(\left(\sqrt{x}\right)^{3} \right)} + 1\right) - \frac{3 \left(\cot^{2}{\left(\left(\sqrt{x}\right)^{3} \right)} + 1\right) \left(18 x^{\frac{3}{2}} \left(\cot^{2}{\left(\left(\sqrt{x}\right)^{3} \right)} + 1\right) + 36 x^{\frac{3}{2}} \cot^{2}{\left(\left(\sqrt{x}\right)^{3} \right)} - 18 \cot{\left(\left(\sqrt{x}\right)^{3} \right)} - \frac{1}{x^{\frac{3}{2}}}\right) \operatorname{asin}^{2}{\left(x \right)}}{8} + 2 \left(\frac{3 x^{2} \operatorname{asin}{\left(x \right)}}{\left(1 - x^{2}\right)^{\frac{5}{2}}} + \frac{3 x}{\left(x^{2} - 1\right)^{2}} + \frac{\operatorname{asin}{\left(x \right)}}{\left(1 - x^{2}\right)^{\frac{3}{2}}}\right) \cot{\left(\left(\sqrt{x}\right)^{3} \right)} + \frac{9 \left(6 x \cot{\left(\left(\sqrt{x}\right)^{3} \right)} - \frac{1}{\sqrt{x}}\right) \left(\cot^{2}{\left(\left(\sqrt{x}\right)^{3} \right)} + 1\right) \operatorname{asin}{\left(x \right)}}{2 \sqrt{1 - x^{2}}}$$
The graph
Derivative of y=ctg((√x)³)×arcsin²x