Производная 5^(tan(x)+log(x))

Вы ввели

 tan(x) + log(x)
5               

$$5^{\log{\left (x \right )} + \tan{\left (x \right )}}$$

Подробное решение

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

2. $\frac{d}{d u} 5^{u} = 5^{u} \log{\left (5 \right )}$

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

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

      1. Есть несколько способов вычислить эту производную.

         Один из способов:

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

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

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

   В результате последовательности правил:

   $5^{\log{\left (x \right )} + \tan{\left (x \right )}} \left(\frac{1}{\cos^{2}{\left (x \right )}} \left(\sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )}\right) + \frac{1}{x}\right) \log{\left (5 \right )}$

4. Теперь упростим:

   $\frac{1}{x} 5^{\log{\left (x \right )} + \tan{\left (x \right )}} \left(\frac{x}{\cos^{2}{\left (x \right )}} + 1\right) \log{\left (5 \right )}$

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Ответ:

$\frac{1}{x} 5^{\log{\left (x \right )} + \tan{\left (x \right )}} \left(\frac{x}{\cos^{2}{\left (x \right )}} + 1\right) \log{\left (5 \right )}$

Первая производная

 tan(x) + log(x) /    1      2   \       
5               *|1 + - + tan (x)|*log(5)
                 \    x          /       

$$5^{\log{\left (x \right )} + \tan{\left (x \right )}} \left(\tan^{2}{\left (x \right )} + 1 + \frac{1}{x}\right) \log{\left (5 \right )}$$

Вторая производная

                 /                        2                                \       
 log(x) + tan(x) |  1    /    1      2   \             /       2   \       |       
5               *|- -- + |1 + - + tan (x)| *log(5) + 2*\1 + tan (x)/*tan(x)|*log(5)
                 |   2   \    x          /                                 |       
                 \  x                                                      /       

$$5^{\log{\left (x \right )} + \tan{\left (x \right )}} \left(2 \left(\tan^{2}{\left (x \right )} + 1\right) \tan{\left (x \right )} + \left(\tan^{2}{\left (x \right )} + 1 + \frac{1}{x}\right)^{2} \log{\left (5 \right )} - \frac{1}{x^{2}}\right) \log{\left (5 \right )}$$

Третья производная

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

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