Производная sec(tan(x)^(-1))

Вы ввели

   /  1   \
sec|------|
   \tan(x)/

$$\sec{\left (\frac{1}{\tan{\left (x \right )}} \right )}$$

Подробное решение

1. Есть несколько способов вычислить эту производную.

   Один из способов:

   1. Заменим $u = \frac{1}{\tan{\left (x \right )}}$.

   2. Производная секанса есть секанс, умноженный на тангенс:

      $\frac{d}{d u} \sec{\left (u \right )} = \tan{\left (u \right )} \sec{\left (u \right )}$

   3. Затем примените цепочку правил. Умножим на $\frac{d}{d x} \frac{1}{\tan{\left (x \right )}}$:

      1. Заменим $u = \tan{\left (x \right )}$.

      2. В силу правила, применим: $\frac{1}{u}$ получим $- \frac{1}{u^{2}}$

      3. Затем примените цепочку правил. Умножим на $\frac{d}{d x} \tan{\left (x \right )}$:

         1. Есть несколько способов вычислить эту производную.

            Один из способов:

            1. $\frac{d}{d x} \tan{\left (x \right )} = \frac{1}{\cos^{2}{\left (x \right )}}$

         В результате последовательности правил:

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

      В результате последовательности правил:

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

2. Теперь упростим:

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

---

Ответ:

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

Первая производная

/        2   \    /  1   \    /  1   \
\-1 - tan (x)/*sec|------|*tan|------|
                  \tan(x)/    \tan(x)/
--------------------------------------
                  2                   
               tan (x)                

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

Вторая производная

              /                     2/  1   \ /       2   \   /       2   \ /       2/  1   \\     /       2   \    /  1   \\            
              |                  tan |------|*\1 + tan (x)/   \1 + tan (x)/*|1 + tan |------||   2*\1 + tan (x)/*tan|------||            
/       2   \ |       /  1   \       \tan(x)/                               \        \tan(x)//                      \tan(x)/|    /  1   \
\1 + tan (x)/*|- 2*tan|------| + -------------------------- + -------------------------------- + ---------------------------|*sec|------|
              |       \tan(x)/               3                               3                                2             |    \tan(x)/
              \                           tan (x)                         tan (x)                          tan (x)          /            
-----------------------------------------------------------------------------------------------------------------------------------------
                                                                  tan(x)                                                                 

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

Третья производная

              /                               2                               2                               2                                     2                                                                                                                                 2                               \            
              |                  /       2   \     3/  1   \     /       2   \     2/  1   \     /       2   \  /       2/  1   \\     /       2   \     /  1   \        2/  1   \ /       2   \     /       2   \ /       2/  1   \\      /       2   \    /  1   \     /       2   \  /       2/  1   \\    /  1   \|            
              |                  \1 + tan (x)/ *tan |------|   6*\1 + tan (x)/ *tan |------|   6*\1 + tan (x)/ *|1 + tan |------||   6*\1 + tan (x)/ *tan|------|   6*tan |------|*\1 + tan (x)/   6*\1 + tan (x)/*|1 + tan |------||   10*\1 + tan (x)/*tan|------|   5*\1 + tan (x)/ *|1 + tan |------||*tan|------||            
/       2   \ |       /  1   \                      \tan(x)/                        \tan(x)/                    \        \tan(x)//                       \tan(x)/         \tan(x)/                                 \        \tan(x)//                       \tan(x)/                    \        \tan(x)//    \tan(x)/|    /  1   \
\1 + tan (x)/*|- 4*tan|------| - --------------------------- - ----------------------------- - ----------------------------------- - ---------------------------- + ---------------------------- + ---------------------------------- + ---------------------------- - -----------------------------------------------|*sec|------|
              |       \tan(x)/                6                              5                                  5                                 4                              3                                 3                                 2                                        6                       |    \tan(x)/
              \                            tan (x)                        tan (x)                            tan (x)                           tan (x)                        tan (x)                           tan (x)                           tan (x)                                  tan (x)                    /            

$$\left(\tan^{2}{\left (x \right )} + 1\right) \left(- \frac{6 \left(\tan^{2}{\left (x \right )} + 1\right)^{2}}{\tan^{5}{\left (x \right )}} \left(\tan^{2}{\left (\frac{1}{\tan{\left (x \right )}} \right )} + 1\right) - \frac{5 \left(\tan^{2}{\left (x \right )} + 1\right)^{2}}{\tan^{6}{\left (x \right )}} \left(\tan^{2}{\left (\frac{1}{\tan{\left (x \right )}} \right )} + 1\right) \tan{\left (\frac{1}{\tan{\left (x \right )}} \right )} - \frac{6 \left(\tan^{2}{\left (x \right )} + 1\right)^{2}}{\tan^{4}{\left (x \right )}} \tan{\left (\frac{1}{\tan{\left (x \right )}} \right )} - \frac{6 \left(\tan^{2}{\left (x \right )} + 1\right)^{2}}{\tan^{5}{\left (x \right )}} \tan^{2}{\left (\frac{1}{\tan{\left (x \right )}} \right )} - \frac{\left(\tan^{2}{\left (x \right )} + 1\right)^{2}}{\tan^{6}{\left (x \right )}} \tan^{3}{\left (\frac{1}{\tan{\left (x \right )}} \right )} + \frac{6}{\tan^{3}{\left (x \right )}} \left(\tan^{2}{\left (x \right )} + 1\right) \left(\tan^{2}{\left (\frac{1}{\tan{\left (x \right )}} \right )} + 1\right) + \frac{10 \tan{\left (\frac{1}{\tan{\left (x \right )}} \right )}}{\tan^{2}{\left (x \right )}} \left(\tan^{2}{\left (x \right )} + 1\right) + \frac{6 \tan^{2}{\left (\frac{1}{\tan{\left (x \right )}} \right )}}{\tan^{3}{\left (x \right )}} \left(\tan^{2}{\left (x \right )} + 1\right) - 4 \tan{\left (\frac{1}{\tan{\left (x \right )}} \right )}\right) \sec{\left (\frac{1}{\tan{\left (x \right )}} \right )}$$