Производная tan(tan(x))^(3)

Вы ввели

   3        
tan (tan(x))

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

Подробное решение

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

2. В силу правила, применим: $u^{3}$ получим $3 u^{2}$

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

   1. Есть несколько способов вычислить эту производную.

      Один из способов:

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

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

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

         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:

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

   В результате последовательности правил:

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

4. Теперь упростим:

   $\frac{3 \tan^{2}{\left (\tan{\left (x \right )} \right )}}{\cos^{2}{\left (x \right )} \cos^{2}{\left (\tan{\left (x \right )} \right )}}$

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Ответ:

$\frac{3 \tan^{2}{\left (\tan{\left (x \right )} \right )}}{\cos^{2}{\left (x \right )} \cos^{2}{\left (\tan{\left (x \right )} \right )}}$

Первая производная

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

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

Вторая производная

  /       2   \ /       2        \ /   2         /       2   \   /       2   \ /       2        \                     \            
6*\1 + tan (x)/*\1 + tan (tan(x))/*\tan (tan(x))*\1 + tan (x)/ + \1 + tan (x)/*\1 + tan (tan(x))/ + tan(x)*tan(tan(x))/*tan(tan(x))

$$6 \left(\tan^{2}{\left (x \right )} + 1\right) \left(\tan^{2}{\left (\tan{\left (x \right )} \right )} + 1\right) \left(\left(\tan^{2}{\left (x \right )} + 1\right) \left(\tan^{2}{\left (\tan{\left (x \right )} \right )} + 1\right) + \left(\tan^{2}{\left (x \right )} + 1\right) \tan^{2}{\left (\tan{\left (x \right )} \right )} + \tan{\left (x \right )} \tan{\left (\tan{\left (x \right )} \right )}\right) \tan{\left (\tan{\left (x \right )} \right )}$$

Третья производная

                                   /             2                   2                                               2                                                                                              2                                                                                        \
  /       2   \ /       2        \ |/       2   \  /       2        \       2         /       2   \     /       2   \     4                2       2                3         /       2   \            /       2   \     2         /       2        \     /       2   \ /       2        \                   |
6*\1 + tan (x)/*\1 + tan (tan(x))/*\\1 + tan (x)/ *\1 + tan (tan(x))/  + tan (tan(x))*\1 + tan (x)/ + 2*\1 + tan (x)/ *tan (tan(x)) + 2*tan (x)*tan (tan(x)) + 6*tan (tan(x))*\1 + tan (x)/*tan(x) + 7*\1 + tan (x)/ *tan (tan(x))*\1 + tan (tan(x))/ + 6*\1 + tan (x)/*\1 + tan (tan(x))/*tan(x)*tan(tan(x))/

$$6 \left(\tan^{2}{\left (x \right )} + 1\right) \left(\tan^{2}{\left (\tan{\left (x \right )} \right )} + 1\right) \left(\left(\tan^{2}{\left (x \right )} + 1\right)^{2} \left(\tan^{2}{\left (\tan{\left (x \right )} \right )} + 1\right)^{2} + 7 \left(\tan^{2}{\left (x \right )} + 1\right)^{2} \left(\tan^{2}{\left (\tan{\left (x \right )} \right )} + 1\right) \tan^{2}{\left (\tan{\left (x \right )} \right )} + 2 \left(\tan^{2}{\left (x \right )} + 1\right)^{2} \tan^{4}{\left (\tan{\left (x \right )} \right )} + 6 \left(\tan^{2}{\left (x \right )} + 1\right) \left(\tan^{2}{\left (\tan{\left (x \right )} \right )} + 1\right) \tan{\left (x \right )} \tan{\left (\tan{\left (x \right )} \right )} + 6 \left(\tan^{2}{\left (x \right )} + 1\right) \tan{\left (x \right )} \tan^{3}{\left (\tan{\left (x \right )} \right )} + \left(\tan^{2}{\left (x \right )} + 1\right) \tan^{2}{\left (\tan{\left (x \right )} \right )} + 2 \tan^{2}{\left (x \right )} \tan^{2}{\left (\tan{\left (x \right )} \right )}\right)$$