Производная log(sin(x))*(tan(x))

Вы ввели

log(sin(x))*tan(x)

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

Подробное решение

1. Применяем правило производной умножения:

   $\frac{d}{d x}\left(f{\left (x \right )} g{\left (x \right )}\right) = f{\left (x \right )} \frac{d}{d x} g{\left (x \right )} + g{\left (x \right )} \frac{d}{d x} f{\left (x \right )}$

   $f{\left (x \right )} = \log{\left (\sin{\left (x \right )} \right )}$; найдём $\frac{d}{d x} f{\left (x \right )}$:

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

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

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

      1. Производная синуса есть косинус:

         $\frac{d}{d x} \sin{\left (x \right )} = \cos{\left (x \right )}$

      В результате последовательности правил:

      $\frac{\cos{\left (x \right )}}{\sin{\left (x \right )}}$

   $g{\left (x \right )} = \tan{\left (x \right )}$; найдём $\frac{d}{d x} g{\left (x \right )}$:

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

2. Теперь упростим:

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

                                2                    2    /       2   \        3                                                                       /       2   \              
          2        /       2   \                3*cos (x)*\1 + tan (x)/   2*cos (x)*tan(x)   2*cos(x)*tan(x)        2    /       2   \               6*\1 + tan (x)/*cos(x)*tan(x)
-3 - 3*tan (x) + 2*\1 + tan (x)/ *log(sin(x)) - ----------------------- + ---------------- + --------------- + 4*tan (x)*\1 + tan (x)/*log(sin(x)) + -----------------------------
                                                           2                     3                sin(x)                                                         sin(x)           
                                                        sin (x)               sin (x)                                                                                             

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