Question

Найдите производную функции:
$\frac{x^{3}}{x^{2} +5}$

Answer 1

Ответ:

$y' = \frac{x^{2}}{ {(x {}^{2} + 5)}}$

Пошаговое объяснение:

$y = \frac{x^{3}}{x^{2} +5} \\ y' = (\frac{x^{3}}{x^{2} +5})' \\ (\frac{u(x)}{v(x)} )' = \frac{u'v - uv'} {v {}^{2} } \\ \\ y' = \frac{(x^{3})' \times (x^{2} +5) - x {}^{3} \times (x {}^{2} + 5)'}{(x^{2} +5) {}^{2} } \\y' = \frac{(3x^{2}) \times (x^{2} +5) - x {}^{3} \times (2x +0)}{ {(x {}^{2} + 5)}^{2} } \\ y' = \frac{3x^{4} + 5x^{2} - 2x {}^{4}}{ {(x {}^{2} + 5)}^{2}} \\ y' = \frac{x^{4} + 5x^{2}}{ {(x {}^{2} + 5)}^{2}} = \frac{x^{2}(x^{2} + 5)}{ {(x {}^{2} + 5)}^{2}} = \frac{x^{2}}{ {(x {}^{2} + 5)}}$