Производная log(cot(x))/log(x)

Вы ввели

log(cot(x))
-----------
   log(x)  

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

Подробное решение

1. Применим правило производной частного:

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

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

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

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

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

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

      1. Есть несколько способов вычислить эту производную.

         Один из способов:

         1. $\frac{d}{d x} \cot{\left (x \right )} = - \frac{1}{\sin^{2}{\left (x \right )}}$

      В результате последовательности правил:

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

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

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

   Теперь применим правило производной деления:

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

2. Теперь упростим:

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

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Ответ:

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

Первая производная

         2                 
 -1 - cot (x)   log(cot(x))
------------- - -----------
cot(x)*log(x)         2    
                 x*log (x) 

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

Вторая производная

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

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

Третья производная

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

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