Производная (3/10)^tan(x)

Вы ввели

    tan(x)
3/10      

$$\left(\frac{3}{10}\right)^{\tan{\left (x \right )}}$$

Подробное решение

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

2. $\frac{d}{d u} \left(\frac{3}{10}\right)^{u} = \left(\frac{3}{10}\right)^{u} \left(- \log{\left (10 \right )} + \log{\left (3 \right )}\right)$

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

   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 последовательности правил:

   $\frac{\left(\frac{3}{10}\right)^{\tan{\left (x \right )}}}{\cos^{2}{\left (x \right )}} \left(- \log{\left (10 \right )} + \log{\left (3 \right )}\right) \left(\sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )}\right)$

4. Теперь упростим:

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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