Производная (tan(x))^x+1

Вы ввели

   x       
tan (x) + 1

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

Подробное решение

1. дифференцируем $\tan^{x}{\left (x \right )} + 1$ почленно:

   1. Не могу найти шаги в поиске этой производной.

      Но производная

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

   2. Производная постоянной $1$ равна нулю.

   В результате: $x^{x} \left(\log{\left (x \right )} + 1\right)$

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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