Производная ((x^2-1)/(x^2-5))^(1/2)

$$sqrt{frac{x^{2} – 1}{x^{2} – 5}}$$

1. Заменим
   u = frac{x^{2} – 1}{x^{2} – 5}
   .

2. В силу правила, применим:
   sqrt{u}
   получим
   frac{1}{2 sqrt{u}}

3. Затем примените цепочку правил. Умножим на
   frac{d}{d x}left(frac{x^{2} – 1}{x^{2} – 5}right)
   :

   1. Применим правило производной частного:

      frac{d}{d x}left(frac{f{left (x right )}}{g{left (x right )}}right) = frac{1}{g^{2}{left (x right )}} left(- f{left (x right )} frac{d}{d x} g{left (x right )} + g{left (x right )} frac{d}{d x} f{left (x right )}right)

      f{left (x right )} = x^{2} – 1
      и
      g{left (x right )} = x^{2} – 5
      $$ .

      Чтобы найти $$
      frac{d}{d x} f{left (x right )}
      :

      1. дифференцируем
         x^{2} – 1
         почленно:

         1. Производная постоянной
            -1
            равна нулю.

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

         В результате:
         2 x

      Чтобы найти
      frac{d}{d x} g{left (x right )}
      :

      1. дифференцируем
         x^{2} – 5
         почленно:

         1. Производная постоянной
            -5
            равна нулю.

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

         В результате:
         2 x

      Теперь применим правило производной деления:

      frac{1}{left(x^{2} – 5right)^{2}} left(2 x left(x^{2} – 5right) – 2 x left(x^{2} – 1right)right)

   В результате последовательности правил:

   frac{2 x left(x^{2} – 5right) – 2 x left(x^{2} – 1right)}{2 sqrt{frac{x^{2} – 1}{x^{2} – 5}} left(x^{2} – 5right)^{2}}

4. Теперь упростим:

   – frac{4 x}{sqrt{frac{x^{2} – 1}{x^{2} – 5}} left(x^{2} – 5right)^{2}}

---

Ответ:

– frac{4 x}{sqrt{frac{x^{2} – 1}{x^{2} – 5}} left(x^{2} – 5right)^{2}}

________
/ 2 / / 2
/ x – 1 / 2 | x x*x – 1/|
/ —— *x – 5/*|—— – ———-|
/ 2 | 2 2 |
/ x – 5 |x – 5 / 2 |
x – 5/ /
———————————————
2
x – 1

$$frac{sqrt{frac{x^{2} – 1}{x^{2} – 5}}}{x^{2} – 1} left(x^{2} – 5right) left(frac{x}{x^{2} – 5} – frac{x left(x^{2} – 1right)}{left(x^{2} – 5right)^{2}}right)$$

Вторая производная

/ 2
| / 2 / 2 / 2 |
| 2 | -1 + x | 2 | -1 + x | 2 | -1 + x | |
_________ | x *|1 – ——-| 2*x *|1 – ——-| 2*x *|1 – ——-| |
/ 2 | 2 2 | 2| | 2| | 2| 2 / 2|
/ -1 + x | -1 + x 4*x -5 + x / -5 + x / -5 + x / 4*x * -1 + x /|
/ ——- *|1 – ——- – ——- + —————– – —————— + —————— + ————–|
/ 2 | 2 2 2 2 2 2 |
/ -5 + x | -5 + x -5 + x -1 + x -1 + x -5 + x / 2 |
-5 + x / /
———————————————————————————————————————-
2
-1 + x

$$frac{sqrt{frac{x^{2} – 1}{x^{2} – 5}}}{x^{2} – 1} left(frac{x^{2}}{x^{2} – 1} left(1 – frac{x^{2} – 1}{x^{2} – 5}right)^{2} – frac{2 x^{2}}{x^{2} – 1} left(1 – frac{x^{2} – 1}{x^{2} – 5}right) + frac{2 x^{2}}{x^{2} – 5} left(1 – frac{x^{2} – 1}{x^{2} – 5}right) – frac{4 x^{2}}{x^{2} – 5} + frac{4 x^{2} left(x^{2} – 1right)}{left(x^{2} – 5right)^{2}} + 1 – frac{x^{2} – 1}{x^{2} – 5}right)$$

Третья производная

/ / 2 2 2 / 2 / 2 2 2 / 2 / 2 2 2 / 2 3 2 / 2 / 2 2 2 / 2 2
| | -1 + x 4*x 4*x * -1 + x /| / 2 / 2 | -1 + x 4*x 4*x * -1 + x /| | -1 + x 2*x 2*x * -1 + x /| / 2 / 2 | -1 + x | | -1 + x 4*x 4*x * -1 + x /| / 2 / 2 / 2 |
| 4*|-1 + ——- + ——- – ————–| | -1 + x | | -1 + x | 4*|-1 + ——- + ——- – ————–| 12*|-1 + ——- + ——- – ————–| 2 | -1 + x | 2 | -1 + x | 3*|1 – ——-|*|-1 + ——- + ——- – ————–| 2 | -1 + x | 2 | -1 + x | 2 | -1 + x | |
_________ | | 2 2 2 | 2*|1 – ——-| 2*|1 – ——-| | 2 2 2 | | 2 2 2 | x *|1 – ——-| 6*x *|1 – ——-| | 2| | 2 2 2 | 8*x *|1 – ——-| 8*x *|1 – ——-| 6*x *|1 – ——-| |
/ 2 | | -5 + x -5 + x / 2 | | 2| | 2| | -5 + x -5 + x / 2 | | -5 + x -5 + x / 2 | | 2| | 2| -5 + x / | -5 + x -5 + x / 2 | | 2| | 2| | 2| |
/ -1 + x | -5 + x / / -5 + x / -5 + x / -5 + x / / -5 + x / / -5 + x / -5 + x / -5 + x / / -5 + x / -5 + x / -5 + x / |
x* / ——- *|- ——————————————- – ————— + ————— + ——————————————- + ——————————————– + —————– – ——————- – ——————————————————— + —————— – ——————- + ——————-|
/ 2 | 2 2 2 2 2 2 2 2 2 / 2 / 2 / 2 / 2|
/ -5 + x | -5 + x -1 + x -5 + x -1 + x -5 + x / 2 / 2 -1 + x / 2 -1 + x /* -5 + x / -1 + x /* -5 + x /|
-1 + x / -1 + x / -1 + x / /
————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————-
2
-1 + x

$$frac{x sqrt{frac{x^{2} – 1}{x^{2} – 5}}}{x^{2} – 1} left(frac{x^{2}}{left(x^{2} – 1right)^{2}} left(1 – frac{x^{2} – 1}{x^{2} – 5}right)^{3} – frac{6 x^{2}}{left(x^{2} – 1right)^{2}} left(1 – frac{x^{2} – 1}{x^{2} – 5}right)^{2} + frac{6 x^{2} left(1 – frac{x^{2} – 1}{x^{2} – 5}right)^{2}}{left(x^{2} – 5right) left(x^{2} – 1right)} + frac{8 x^{2}}{left(x^{2} – 1right)^{2}} left(1 – frac{x^{2} – 1}{x^{2} – 5}right) – frac{8 x^{2} left(1 – frac{x^{2} – 1}{x^{2} – 5}right)}{left(x^{2} – 5right) left(x^{2} – 1right)} – frac{3}{x^{2} – 1} left(1 – frac{x^{2} – 1}{x^{2} – 5}right) left(frac{4 x^{2}}{x^{2} – 5} – frac{4 x^{2} left(x^{2} – 1right)}{left(x^{2} – 5right)^{2}} – 1 + frac{x^{2} – 1}{x^{2} – 5}right) – frac{1}{x^{2} – 1} left(2 – frac{2 x^{2} – 2}{x^{2} – 5}right) + frac{1}{x^{2} – 5} left(2 – frac{2 x^{2} – 2}{x^{2} – 5}right) + frac{1}{x^{2} – 1} left(frac{16 x^{2}}{x^{2} – 5} – frac{16 x^{2} left(x^{2} – 1right)}{left(x^{2} – 5right)^{2}} – 4 + frac{4 x^{2} – 4}{x^{2} – 5}right) + frac{1}{x^{2} – 5} left(frac{24 x^{2}}{x^{2} – 5} – frac{24 x^{2} left(x^{2} – 1right)}{left(x^{2} – 5right)^{2}} – 12 + frac{12 x^{2} – 12}{x^{2} – 5}right) – frac{1}{x^{2} – 5} left(frac{16 x^{2}}{x^{2} – 5} – frac{16 x^{2} left(x^{2} – 1right)}{left(x^{2} – 5right)^{2}} – 4 + frac{4 x^{2} – 4}{x^{2} – 5}right)right)$$