Производная sec(sqrt(tan(x)))

Вы ввели

   /  ________\
sec\\/ tan(x) /

$$\sec{\left (\sqrt{\tan{\left (x \right )}} \right )}$$

Подробное решение

1. Есть несколько способов вычислить эту производную.

   Один из способов:

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

   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 $\frac{d}{d x} \sqrt{\tan{\left (x \right )}}$:

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

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

      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:

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

      В результате последовательности правил:

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

2. Теперь упростим:

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

              /                             /       2   \    /  ________\      2/  ________\ /       2   \   /       2   \ /       2/  ________\\\                
/       2   \ |  ________    /  ________\   \1 + tan (x)/*tan\\/ tan(x) /   tan \\/ tan(x) /*\1 + tan (x)/   \1 + tan (x)/*\1 + tan \\/ tan(x) //|    /  ________\
\1 + tan (x)/*|\/ tan(x) *tan\\/ tan(x) / - ----------------------------- + ------------------------------ + ------------------------------------|*sec\\/ tan(x) /
              |                                           3/2                          4*tan(x)                            4*tan(x)              |                
              \                                      4*tan   (x)                                                                                 /                

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

Третья производная

              /                                                                                                                         2                                   2                                                                       2                                   2                                  2                                       \                
              |                                   2/  ________\ /       2   \     /       2   \ /       2/  ________\\     /       2   \     2/  ________\     /       2   \  /       2/  ________\\   /       2   \    /  ________\   /       2   \     3/  ________\     /       2   \     /  ________\     /       2   \  /       2/  ________\\    /  ________\|                
/       2   \ |     3/2       /  ________\   3*tan \\/ tan(x) /*\1 + tan (x)/   3*\1 + tan (x)/*\1 + tan \\/ tan(x) //   3*\1 + tan (x)/ *tan \\/ tan(x) /   3*\1 + tan (x)/ *\1 + tan \\/ tan(x) //   \1 + tan (x)/*tan\\/ tan(x) /   \1 + tan (x)/ *tan \\/ tan(x) /   3*\1 + tan (x)/ *tan\\/ tan(x) /   5*\1 + tan (x)/ *\1 + tan \\/ tan(x) //*tan\\/ tan(x) /|    /  ________\
\1 + tan (x)/*|2*tan   (x)*tan\\/ tan(x) / + -------------------------------- + -------------------------------------- - --------------------------------- - --------------------------------------- - ----------------------------- + ------------------------------- + -------------------------------- + -------------------------------------------------------|*sec\\/ tan(x) /
              |                                             2                                     2                                       2                                      2                                  ________                          3/2                               5/2                                            3/2                         |                
              \                                                                                                                      8*tan (x)                              8*tan (x)                           2*\/ tan(x)                      8*tan   (x)                       8*tan   (x)                                    8*tan   (x)                      /                

$$\left(\tan^{2}{\left (x \right )} + 1\right) \left(- \frac{3 \left(\tan^{2}{\left (x \right )} + 1\right)^{2}}{8 \tan^{2}{\left (x \right )}} \left(\tan^{2}{\left (\sqrt{\tan{\left (x \right )}} \right )} + 1\right) + \frac{5 \left(\tan^{2}{\left (x \right )} + 1\right)^{2}}{8 \tan^{\frac{3}{2}}{\left (x \right )}} \left(\tan^{2}{\left (\sqrt{\tan{\left (x \right )}} \right )} + 1\right) \tan{\left (\sqrt{\tan{\left (x \right )}} \right )} - \frac{3 \left(\tan^{2}{\left (x \right )} + 1\right)^{2}}{8 \tan^{2}{\left (x \right )}} \tan^{2}{\left (\sqrt{\tan{\left (x \right )}} \right )} + \frac{\left(\tan^{2}{\left (x \right )} + 1\right)^{2}}{8 \tan^{\frac{3}{2}}{\left (x \right )}} \tan^{3}{\left (\sqrt{\tan{\left (x \right )}} \right )} + \frac{3 \left(\tan^{2}{\left (x \right )} + 1\right)^{2}}{8 \tan^{\frac{5}{2}}{\left (x \right )}} \tan{\left (\sqrt{\tan{\left (x \right )}} \right )} + \frac{3}{2} \left(\tan^{2}{\left (x \right )} + 1\right) \left(\tan^{2}{\left (\sqrt{\tan{\left (x \right )}} \right )} + 1\right) + \frac{3}{2} \left(\tan^{2}{\left (x \right )} + 1\right) \tan^{2}{\left (\sqrt{\tan{\left (x \right )}} \right )} - \frac{\tan{\left (\sqrt{\tan{\left (x \right )}} \right )}}{2 \sqrt{\tan{\left (x \right )}}} \left(\tan^{2}{\left (x \right )} + 1\right) + 2 \tan^{\frac{3}{2}}{\left (x \right )} \tan{\left (\sqrt{\tan{\left (x \right )}} \right )}\right) \sec{\left (\sqrt{\tan{\left (x \right )}} \right )}$$