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Derivative of arctg((tgx-ctgx)/sqrt(2)(2))

Function f() - derivative -N order at the point
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    /tan(x) - cot(x)  \
atan|---------------*2|
    |       ___       |
    \     \/ 2        /
atan(2tan(x)cot(x)2)\operatorname{atan}{\left(2 \frac{\tan{\left(x \right)} - \cot{\left(x \right)}}{\sqrt{2}} \right)}
atan(((tan(x) - cot(x))/sqrt(2))*2)
The graph
02468-8-6-4-2-1010-1010
The first derivative [src]
  ___ /       2         2   \
\/ 2 *\2 + cot (x) + tan (x)/
-----------------------------
                          2  
   1 + 2*(tan(x) - cot(x))   
2(tan2(x)+cot2(x)+2)2(tan(x)cot(x))2+1\frac{\sqrt{2} \left(\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)} + 2\right)}{2 \left(\tan{\left(x \right)} - \cot{\left(x \right)}\right)^{2} + 1}
The second derivative [src]
        /                                                                       2                   \
        |                                                /       2         2   \                    |
    ___ |/       2   \          /       2   \          2*\2 + cot (x) + tan (x)/ *(-cot(x) + tan(x))|
2*\/ 2 *|\1 + tan (x)/*tan(x) - \1 + cot (x)/*cot(x) - ---------------------------------------------|
        |                                                                                2          |
        \                                                        1 + 2*(-cot(x) + tan(x))           /
-----------------------------------------------------------------------------------------------------
                                                              2                                      
                                      1 + 2*(-cot(x) + tan(x))                                       
22((tan2(x)+1)tan(x)(cot2(x)+1)cot(x)2(tan(x)cot(x))(tan2(x)+cot2(x)+2)22(tan(x)cot(x))2+1)2(tan(x)cot(x))2+1\frac{2 \sqrt{2} \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} - \frac{2 \left(\tan{\left(x \right)} - \cot{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)} + 2\right)^{2}}{2 \left(\tan{\left(x \right)} - \cot{\left(x \right)}\right)^{2} + 1}\right)}{2 \left(\tan{\left(x \right)} - \cot{\left(x \right)}\right)^{2} + 1}
The third derivative [src]
        /                                                           3                                                                                                     3                                                                                              \
        |             2                2     /       2         2   \                                                                             2 /       2         2   \                          //       2   \          /       2   \       \ /       2         2   \|
    ___ |/       2   \    /       2   \    2*\2 + cot (x) + tan (x)/         2    /       2   \        2    /       2   \   16*(-cot(x) + tan(x)) *\2 + cot (x) + tan (x)/    12*(-cot(x) + tan(x))*\\1 + tan (x)/*tan(x) - \1 + cot (x)/*cot(x)/*\2 + cot (x) + tan (x)/|
2*\/ 2 *|\1 + cot (x)/  + \1 + tan (x)/  - -------------------------- + 2*cot (x)*\1 + cot (x)/ + 2*tan (x)*\1 + tan (x)/ + ----------------------------------------------- - -------------------------------------------------------------------------------------------|
        |                                                          2                                                                                             2                                                                     2                                 |
        |                                  1 + 2*(-cot(x) + tan(x))                                                                   /                        2\                                              1 + 2*(-cot(x) + tan(x))                                  |
        \                                                                                                                             \1 + 2*(-cot(x) + tan(x)) /                                                                                                        /
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                                                                                                                                                2                                                                                                                         
                                                                                                                        1 + 2*(-cot(x) + tan(x))                                                                                                                          
22(12((tan2(x)+1)tan(x)(cot2(x)+1)cot(x))(tan(x)cot(x))(tan2(x)+cot2(x)+2)2(tan(x)cot(x))2+1+(tan2(x)+1)2+2(tan2(x)+1)tan2(x)+(cot2(x)+1)2+2(cot2(x)+1)cot2(x)2(tan2(x)+cot2(x)+2)32(tan(x)cot(x))2+1+16(tan(x)cot(x))2(tan2(x)+cot2(x)+2)3(2(tan(x)cot(x))2+1)2)2(tan(x)cot(x))2+1\frac{2 \sqrt{2} \left(- \frac{12 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}\right) \left(\tan{\left(x \right)} - \cot{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)} + 2\right)}{2 \left(\tan{\left(x \right)} - \cot{\left(x \right)}\right)^{2} + 1} + \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + \left(\cot^{2}{\left(x \right)} + 1\right)^{2} + 2 \left(\cot^{2}{\left(x \right)} + 1\right) \cot^{2}{\left(x \right)} - \frac{2 \left(\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)} + 2\right)^{3}}{2 \left(\tan{\left(x \right)} - \cot{\left(x \right)}\right)^{2} + 1} + \frac{16 \left(\tan{\left(x \right)} - \cot{\left(x \right)}\right)^{2} \left(\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)} + 2\right)^{3}}{\left(2 \left(\tan{\left(x \right)} - \cot{\left(x \right)}\right)^{2} + 1\right)^{2}}\right)}{2 \left(\tan{\left(x \right)} - \cot{\left(x \right)}\right)^{2} + 1}