Производная (sec(3^cot(x)))^2

Вы ввели

   2/ cot(x)\
sec \3      /

$$\sec^{2}{\left (3^{\cot{\left (x \right )}} \right )}$$

Подробное решение

1. Заменим $u = \sec{\left (3^{\cot{\left (x \right )}} \right )}$.

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

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

   1. Есть несколько способов вычислить эту производную.

      Один из способов:

      1. Заменим $u = 3^{\cot{\left (x \right )}}$.

      2. Производная секанса есть секанс, умноженный на тангенс:

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

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

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

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

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

            1. Есть несколько способов вычислить эту производную.

               Один из способов:

               1. $\frac{d}{d x} \cot{\left (x \right )} = - \frac{1}{\sin^{2}{\left (x \right )}}$

            В результате последовательности правил:

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

         В результате последовательности правил:

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

   В результате последовательности правил:

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

4. Теперь упростим:

   $- \frac{3^{\frac{1}{\tan{\left (x \right )}}} \log{\left (9 \right )} \sin{\left (3^{\frac{1}{\tan{\left (x \right )}}} \right )}}{\sin^{2}{\left (x \right )} \cos^{3}{\left (3^{\frac{1}{\tan{\left (x \right )}}} \right )}}$

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Ответ:

$- \frac{3^{\frac{1}{\tan{\left (x \right )}}} \log{\left (9 \right )} \sin{\left (3^{\frac{1}{\tan{\left (x \right )}}} \right )}}{\sin^{2}{\left (x \right )} \cos^{3}{\left (3^{\frac{1}{\tan{\left (x \right )}}} \right )}}$

Первая производная

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

$$2 \cdot 3^{\cot{\left (x \right )}} \left(- \cot^{2}{\left (x \right )} - 1\right) \log{\left (3 \right )} \tan{\left (3^{\cot{\left (x \right )}} \right )} \sec^{2}{\left (3^{\cot{\left (x \right )}} \right )}$$

Вторая производная

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

$$2 \cdot 3^{\cot{\left (x \right )}} \left(\cot^{2}{\left (x \right )} + 1\right) \left(3^{\cot{\left (x \right )}} \left(\tan^{2}{\left (3^{\cot{\left (x \right )}} \right )} + 1\right) \left(\cot^{2}{\left (x \right )} + 1\right) \log{\left (3 \right )} + 2 \cdot 3^{\cot{\left (x \right )}} \left(\cot^{2}{\left (x \right )} + 1\right) \log{\left (3 \right )} \tan^{2}{\left (3^{\cot{\left (x \right )}} \right )} + \left(\cot^{2}{\left (x \right )} + 1\right) \log{\left (3 \right )} \tan{\left (3^{\cot{\left (x \right )}} \right )} + 2 \tan{\left (3^{\cot{\left (x \right )}} \right )} \cot{\left (x \right )}\right) \log{\left (3 \right )} \sec^{2}{\left (3^{\cot{\left (x \right )}} \right )}$$

Третья производная

                                       /                                                                     2                                               2                                                        2                                                2                                                                                                                                                           2                                                                                                \       
    cot(x)    2/ cot(x)\ /       2   \ |  /       2   \    / cot(x)\        2       / cot(x)\   /       2   \     2       / cot(x)\      cot(x) /       2   \     2    /       2/ cot(x)\\      2*cot(x) /       2   \     2       3/ cot(x)\      cot(x) /       2   \     2       2/ cot(x)\     /       2   \                  / cot(x)\      cot(x) /       2   \ /       2/ cot(x)\\                    2*cot(x) /       2   \     2    /       2/ cot(x)\\    / cot(x)\       cot(x)    2/ cot(x)\ /       2   \              |       
-2*3      *sec \3      /*\1 + cot (x)/*\2*\1 + cot (x)/*tan\3      / + 4*cot (x)*tan\3      / + \1 + cot (x)/ *log (3)*tan\3      / + 3*3      *\1 + cot (x)/ *log (3)*\1 + tan \3      // + 4*3        *\1 + cot (x)/ *log (3)*tan \3      / + 6*3      *\1 + cot (x)/ *log (3)*tan \3      / + 6*\1 + cot (x)/*cot(x)*log(3)*tan\3      / + 6*3      *\1 + cot (x)/*\1 + tan \3      //*cot(x)*log(3) + 8*3        *\1 + cot (x)/ *log (3)*\1 + tan \3      //*tan\3      / + 12*3      *tan \3      /*\1 + cot (x)/*cot(x)*log(3)/*log(3)

$$- 2 \cdot 3^{\cot{\left (x \right )}} \left(\cot^{2}{\left (x \right )} + 1\right) \left(8 \cdot 3^{2 \cot{\left (x \right )}} \left(\tan^{2}{\left (3^{\cot{\left (x \right )}} \right )} + 1\right) \left(\cot^{2}{\left (x \right )} + 1\right)^{2} \log^{2}{\left (3 \right )} \tan{\left (3^{\cot{\left (x \right )}} \right )} + 4 \cdot 3^{2 \cot{\left (x \right )}} \left(\cot^{2}{\left (x \right )} + 1\right)^{2} \log^{2}{\left (3 \right )} \tan^{3}{\left (3^{\cot{\left (x \right )}} \right )} + 3 \cdot 3^{\cot{\left (x \right )}} \left(\tan^{2}{\left (3^{\cot{\left (x \right )}} \right )} + 1\right) \left(\cot^{2}{\left (x \right )} + 1\right)^{2} \log^{2}{\left (3 \right )} + 6 \cdot 3^{\cot{\left (x \right )}} \left(\tan^{2}{\left (3^{\cot{\left (x \right )}} \right )} + 1\right) \left(\cot^{2}{\left (x \right )} + 1\right) \log{\left (3 \right )} \cot{\left (x \right )} + 6 \cdot 3^{\cot{\left (x \right )}} \left(\cot^{2}{\left (x \right )} + 1\right)^{2} \log^{2}{\left (3 \right )} \tan^{2}{\left (3^{\cot{\left (x \right )}} \right )} + 12 \cdot 3^{\cot{\left (x \right )}} \left(\cot^{2}{\left (x \right )} + 1\right) \log{\left (3 \right )} \tan^{2}{\left (3^{\cot{\left (x \right )}} \right )} \cot{\left (x \right )} + \left(\cot^{2}{\left (x \right )} + 1\right)^{2} \log^{2}{\left (3 \right )} \tan{\left (3^{\cot{\left (x \right )}} \right )} + 6 \left(\cot^{2}{\left (x \right )} + 1\right) \log{\left (3 \right )} \tan{\left (3^{\cot{\left (x \right )}} \right )} \cot{\left (x \right )} + 2 \left(\cot^{2}{\left (x \right )} + 1\right) \tan{\left (3^{\cot{\left (x \right )}} \right )} + 4 \tan{\left (3^{\cot{\left (x \right )}} \right )} \cot^{2}{\left (x \right )}\right) \log{\left (3 \right )} \sec^{2}{\left (3^{\cot{\left (x \right )}} \right )}$$