Производная 3^x/tan(x)

Вы ввели

   x  
  3   
------
tan(x)

$$\frac{3^{x}}{\tan{\left (x \right )}}$$

Подробное решение

1. Применим правило производной частного:

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

   $f{\left (x \right )} = 3^{x}$ и $g{\left (x \right )} = \tan{\left (x \right )}$.

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

   1. $\frac{d}{d x} 3^{x} = 3^{x} \log{\left (3 \right )}$

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

   1. Есть несколько способов вычислить эту производную.

      Один из способов:

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

   Теперь применим правило производной деления:

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

2. Теперь упростим:

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

   /                                          3                   2                                                                     2       \
   |                    3        /       2   \       /       2   \      /       2   \               2    /       2   \     /       2   \        |
 x |          2      log (3)   6*\1 + tan (x)/    10*\1 + tan (x)/    6*\1 + tan (x)/*log(3)   3*log (3)*\1 + tan (x)/   6*\1 + tan (x)/ *log(3)|
3 *|-4 - 4*tan (x) + ------- - ---------------- + ----------------- - ---------------------- - ----------------------- + -----------------------|
   |                  tan(x)          4                   2                   tan(x)                      2                         3           |
   \                               tan (x)             tan (x)                                         tan (x)                   tan (x)        /

$$3^{x} \left(- \frac{6 \left(\tan^{2}{\left (x \right )} + 1\right)^{3}}{\tan^{4}{\left (x \right )}} + \frac{10 \left(\tan^{2}{\left (x \right )} + 1\right)^{2}}{\tan^{2}{\left (x \right )}} + \frac{6 \left(\tan^{2}{\left (x \right )} + 1\right)^{2}}{\tan^{3}{\left (x \right )}} \log{\left (3 \right )} - \frac{6 \log{\left (3 \right )}}{\tan{\left (x \right )}} \left(\tan^{2}{\left (x \right )} + 1\right) - \frac{3 \log^{2}{\left (3 \right )}}{\tan^{2}{\left (x \right )}} \left(\tan^{2}{\left (x \right )} + 1\right) - 4 \tan^{2}{\left (x \right )} - 4 + \frac{\log^{3}{\left (3 \right )}}{\tan{\left (x \right )}}\right)$$