Производная (sin(x))^(2-tan(x))

Вы ввели

   2 - tan(x)   
sin          (x)

$$\sin^{- \tan{\left (x \right )} + 2}{\left (x \right )}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

   $\left(\log{\left (- \tan{\left (x \right )} + 2 \right )} + 1\right) \left(- \tan{\left (x \right )} + 2\right)^{- \tan{\left (x \right )} + 2}$

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Ответ:

$\left(\log{\left (- \tan{\left (x \right )} + 2 \right )} + 1\right) \left(- \tan{\left (x \right )} + 2\right)^{- \tan{\left (x \right )} + 2}$

Первая производная

   2 - tan(x)    //        2   \               (2 - tan(x))*cos(x)\
sin          (x)*|\-1 - tan (x)/*log(sin(x)) + -------------------|
                 \                                    sin(x)      /

$$\left(\frac{\cos{\left (x \right )}}{\sin{\left (x \right )}} \left(- \tan{\left (x \right )} + 2\right) + \left(- \tan^{2}{\left (x \right )} - 1\right) \log{\left (\sin{\left (x \right )} \right )}\right) \sin^{- \tan{\left (x \right )} + 2}{\left (x \right )}$$

Вторая производная

                 /                                                       2      2                      /       2   \                                                     \
   2 - tan(x)    |     //       2   \               (-2 + tan(x))*cos(x)\    cos (x)*(-2 + tan(x))   2*\1 + tan (x)/*cos(x)     /       2   \                            |
sin          (x)*|-2 + |\1 + tan (x)/*log(sin(x)) + --------------------|  + --------------------- - ---------------------- - 2*\1 + tan (x)/*log(sin(x))*tan(x) + tan(x)|
                 |     \                                   sin(x)       /              2                     sin(x)                                                      |
                 \                                                                  sin (x)                                                                              /

$$\left(\left(\frac{\cos{\left (x \right )}}{\sin{\left (x \right )}} \left(\tan{\left (x \right )} - 2\right) + \left(\tan^{2}{\left (x \right )} + 1\right) \log{\left (\sin{\left (x \right )} \right )}\right)^{2} + \frac{\cos^{2}{\left (x \right )}}{\sin^{2}{\left (x \right )}} \left(\tan{\left (x \right )} - 2\right) - 2 \left(\tan^{2}{\left (x \right )} + 1\right) \log{\left (\sin{\left (x \right )} \right )} \tan{\left (x \right )} - \frac{2 \cos{\left (x \right )}}{\sin{\left (x \right )}} \left(\tan^{2}{\left (x \right )} + 1\right) + \tan{\left (x \right )} - 2\right) \sin^{- \tan{\left (x \right )} + 2}{\left (x \right )}$$

Третья производная

                 /                                                      3                              2                                                                    /                2                      /       2   \                                            \                                              3                                                  2    /       2   \     /       2   \              \
   2 - tan(x)    |    //       2   \               (-2 + tan(x))*cos(x)\         2        /       2   \                  //       2   \               (-2 + tan(x))*cos(x)\ |             cos (x)*(-2 + tan(x))   2*\1 + tan (x)/*cos(x)     /       2   \                   |        2    /       2   \               2*cos (x)*(-2 + tan(x))   2*(-2 + tan(x))*cos(x)   3*cos (x)*\1 + tan (x)/   6*\1 + tan (x)/*cos(x)*tan(x)|
sin          (x)*|3 - |\1 + tan (x)/*log(sin(x)) + --------------------|  + 3*tan (x) - 2*\1 + tan (x)/ *log(sin(x)) + 3*|\1 + tan (x)/*log(sin(x)) + --------------------|*|2 - tan(x) - --------------------- + ---------------------- + 2*\1 + tan (x)/*log(sin(x))*tan(x)| - 4*tan (x)*\1 + tan (x)/*log(sin(x)) - ----------------------- - ---------------------- + ----------------------- - -----------------------------|
                 |    \                                   sin(x)       /                                                 \                                   sin(x)       / |                       2                     sin(x)                                             |                                                    3                      sin(x)                      2                          sin(x)           |
                 \                                                                                                                                                          \                    sin (x)                                                                     /                                                 sin (x)                                            sin (x)                                        /

$$\left(- \left(\frac{\cos{\left (x \right )}}{\sin{\left (x \right )}} \left(\tan{\left (x \right )} - 2\right) + \left(\tan^{2}{\left (x \right )} + 1\right) \log{\left (\sin{\left (x \right )} \right )}\right)^{3} + 3 \left(\frac{\cos{\left (x \right )}}{\sin{\left (x \right )}} \left(\tan{\left (x \right )} - 2\right) + \left(\tan^{2}{\left (x \right )} + 1\right) \log{\left (\sin{\left (x \right )} \right )}\right) \left(- \frac{\cos^{2}{\left (x \right )}}{\sin^{2}{\left (x \right )}} \left(\tan{\left (x \right )} - 2\right) + 2 \left(\tan^{2}{\left (x \right )} + 1\right) \log{\left (\sin{\left (x \right )} \right )} \tan{\left (x \right )} + \frac{2 \cos{\left (x \right )}}{\sin{\left (x \right )}} \left(\tan^{2}{\left (x \right )} + 1\right) - \tan{\left (x \right )} + 2\right) - \frac{2 \cos{\left (x \right )}}{\sin{\left (x \right )}} \left(\tan{\left (x \right )} - 2\right) - \frac{2 \cos^{3}{\left (x \right )}}{\sin^{3}{\left (x \right )}} \left(\tan{\left (x \right )} - 2\right) - 2 \left(\tan^{2}{\left (x \right )} + 1\right)^{2} \log{\left (\sin{\left (x \right )} \right )} - 4 \left(\tan^{2}{\left (x \right )} + 1\right) \log{\left (\sin{\left (x \right )} \right )} \tan^{2}{\left (x \right )} - \frac{6 \cos{\left (x \right )}}{\sin{\left (x \right )}} \left(\tan^{2}{\left (x \right )} + 1\right) \tan{\left (x \right )} + \frac{3 \cos^{2}{\left (x \right )}}{\sin^{2}{\left (x \right )}} \left(\tan^{2}{\left (x \right )} + 1\right) + 3 \tan^{2}{\left (x \right )} + 3\right) \sin^{- \tan{\left (x \right )} + 2}{\left (x \right )}$$