Производная log(tan(sec(x)+tan(x)))

Вы ввели

log(tan(sec(x) + tan(x)))

$$\log{\left (\tan{\left (\tan{\left (x \right )} + \sec{\left (x \right )} \right )} \right )}$$

Подробное решение

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

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

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

   1. Есть несколько способов вычислить эту производную.

      Один из способов:

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

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

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

         1. дифференцируем $\tan{\left (x \right )} + \sec{\left (x \right )}$ почленно:

            1. Есть несколько способов вычислить эту производную.

               Один из способов:

               1. Производная секанса есть секанс, умноженный на тангенс:

                  $\frac{d}{d x} \sec{\left (x \right )} = \tan{\left (x \right )} \sec{\left (x \right )}$

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

         В результате последовательности правил:

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

   В результате последовательности правил:

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

4. Теперь упростим:

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

                            /               2                                                                                                                                                                                                                                                                        3                                                            2                              3                                                                                                                               \
                            |  /       2   \       3                  2    /       2   \     /       2   \                                                3                                                                                                                             /       2                   \  /       2                 \     /       2                 \  /       2                   \      /       2                 \ /       2                   \ /   2             /       2   \            /       2   \       \|
/       2                 \ |2*\1 + tan (x)/  + tan (x)*sec(x) + 4*tan (x)*\1 + tan (x)/ + 5*\1 + tan (x)/*sec(x)*tan(x)     /       2                   \                           /       2                   \ /   2             /       2   \            /       2   \       \   4*\1 + tan (x) + sec(x)*tan(x)/ *\1 + tan (sec(x) + tan(x))/   2*\1 + tan (sec(x) + tan(x))/ *\1 + tan (x) + sec(x)*tan(x)/    3*\1 + tan (sec(x) + tan(x))/*\1 + tan (x) + sec(x)*tan(x)/*\tan (x)*sec(x) + \1 + tan (x)/*sec(x) + 2*\1 + tan (x)/*tan(x)/|
\1 + tan (sec(x) + tan(x))/*|------------------------------------------------------------------------------------------- + 4*\1 + tan (x) + sec(x)*tan(x)/ *tan(sec(x) + tan(x)) + 6*\1 + tan (x) + sec(x)*tan(x)/*\tan (x)*sec(x) + \1 + tan (x)/*sec(x) + 2*\1 + tan (x)/*tan(x)/ - ------------------------------------------------------------ + ------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------|
                            |                                    tan(sec(x) + tan(x))                                                                                                                                                                                                                     tan(sec(x) + tan(x))                                              3                                                                                              2                                                                     |
                            \                                                                                                                                                                                                                                                                                                                                            tan (sec(x) + tan(x))                                                                          tan (sec(x) + tan(x))                                                    /

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