Производная sec(tan(x))

Вы ввели

sec(tan(x))

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

Подробное решение

1. Есть несколько способов вычислить эту производную.

   Один из способов:

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

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

      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:

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

2. Теперь упростим:

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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

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