Производная log(tan(x)/x)

Вы ввели

   /tan(x)\
log|------|
   \  x   /

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

Подробное решение

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

2. Производная $\log{\left (u \right )}$ является $\frac{1}{u}$.

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

   1. Применим правило производной частного:

      $\frac{d}{d x}\left(\frac{f{\left (x \right )}}{g{\left (x \right )}}\right) = \frac{1}{g^{2}{\left (x \right )}} \left(- f{\left (x \right )} \frac{d}{d x} g{\left (x \right )} + g{\left (x \right )} \frac{d}{d x} f{\left (x \right )}\right)$

      $f{\left (x \right )} = \tan{\left (x \right )}$ и $g{\left (x \right )} = x$.

      Чтобы найти $\frac{d}{d x} f{\left (x \right )}$:

      1. Есть несколько способов вычислить эту производную.

         Один из способов:

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

      Чтобы найти $\frac{d}{d x} g{\left (x \right )}$:

      1. В силу правила, применим: $x$ получим $1$

      Теперь применим правило производной деления:

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

   В результате последовательности правил:

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

4. Теперь упростим:

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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