Производная (x)/log(tan(x))

Вы ввели

     x     
-----------
log(tan(x))

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

Подробное решение

1. Применим правило производной частного:

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

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

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

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

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

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

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

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

      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:

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

   Теперь применим правило производной деления:

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

2. Теперь упростим:

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

              /                                                     2                                                               2                     2                      \
              |                    /       2   \       /       2   \        /       2   \       /       2   \          /       2   \         /       2   \          /       2   \|
/       2   \ |                  3*\1 + tan (x)/   2*x*\1 + tan (x)/    4*x*\1 + tan (x)/     6*\1 + tan (x)/      6*x*\1 + tan (x)/     6*x*\1 + tan (x)/     12*x*\1 + tan (x)/|
\1 + tan (x)/*|-6 - 4*x*tan(x) + --------------- - ------------------ + ----------------- + ------------------- - ------------------- - -------------------- + ------------------|
              |                         2                  3                  tan(x)                       2                     3         2            3      log(tan(x))*tan(x)|
              \                      tan (x)            tan (x)                             log(tan(x))*tan (x)   log(tan(x))*tan (x)   log (tan(x))*tan (x)                     /
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                      2                                                                                           
                                                                                   log (tan(x))                                                                                   

$$\frac{1}{\log^{2}{\left (\tan{\left (x \right )} \right )}} \left(\tan^{2}{\left (x \right )} + 1\right) \left(- \frac{2 x \left(\tan^{2}{\left (x \right )} + 1\right)^{2}}{\tan^{3}{\left (x \right )}} - \frac{6 x \left(\tan^{2}{\left (x \right )} + 1\right)^{2}}{\log{\left (\tan{\left (x \right )} \right )} \tan^{3}{\left (x \right )}} - \frac{6 x \left(\tan^{2}{\left (x \right )} + 1\right)^{2}}{\log^{2}{\left (\tan{\left (x \right )} \right )} \tan^{3}{\left (x \right )}} + \frac{4 x \left(\tan^{2}{\left (x \right )} + 1\right)}{\tan{\left (x \right )}} + \frac{12 x \left(\tan^{2}{\left (x \right )} + 1\right)}{\log{\left (\tan{\left (x \right )} \right )} \tan{\left (x \right )}} - 4 x \tan{\left (x \right )} + \frac{3 \tan^{2}{\left (x \right )} + 3}{\tan^{2}{\left (x \right )}} + \frac{6 \tan^{2}{\left (x \right )} + 6}{\log{\left (\tan{\left (x \right )} \right )} \tan^{2}{\left (x \right )}} - 6\right)$$