Производная tan(x)/sin(x)

Вы ввели

tan(x)
------
sin(x)

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

Подробное решение

1. Применим правило производной частного:

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

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

   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. Производная синуса есть косинус:

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

   Теперь применим правило производной деления:

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

2. Теперь упростим:

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

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Ответ:

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

Первая производная

       2                   
1 + tan (x)   cos(x)*tan(x)
----------- - -------------
   sin(x)           2      
                 sin (x)   

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

Вторая производная

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

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

Третья производная

                   2                                              3                                    2    /       2   \     /       2   \              
      /       2   \         2           2    /       2   \   6*cos (x)*tan(x)   5*cos(x)*tan(x)   6*cos (x)*\1 + tan (x)/   6*\1 + tan (x)/*cos(x)*tan(x)
3 + 2*\1 + tan (x)/  + 3*tan (x) + 4*tan (x)*\1 + tan (x)/ - ---------------- - --------------- + ----------------------- - -----------------------------
                                                                    3                sin(x)                  2                          sin(x)           
                                                                 sin (x)                                  sin (x)                                        
---------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                          sin(x)                                                                         

$$\frac{1}{\sin{\left (x \right )}} \left(2 \left(\tan^{2}{\left (x \right )} + 1\right)^{2} + 4 \left(\tan^{2}{\left (x \right )} + 1\right) \tan^{2}{\left (x \right )} - \frac{6 \cos{\left (x \right )}}{\sin{\left (x \right )}} \left(\tan^{2}{\left (x \right )} + 1\right) \tan{\left (x \right )} + \frac{6 \cos^{2}{\left (x \right )}}{\sin^{2}{\left (x \right )}} \left(\tan^{2}{\left (x \right )} + 1\right) + 3 \tan^{2}{\left (x \right )} + 3 - \frac{5 \cos{\left (x \right )}}{\sin{\left (x \right )}} \tan{\left (x \right )} - \frac{6 \cos^{3}{\left (x \right )}}{\sin^{3}{\left (x \right )}} \tan{\left (x \right )}\right)$$