Question

Найти производнве dy/dx

Answer 1

Ответ:

Формула:   $\Big(\dfrac{u}{v}\Big)'=\dfrac{u'v-uv'}{v^2}$   .

$\displaystyle y=\frac{3\, \sqrt[3]{x^5+5x^4-5}}{x}\\\\\\y'=\frac{dy}{dx}=\frac{3\cdot \frac{1}{3}\, (x^5+5x^4-5)^{-\frac{2}{3}}\cdot (5x^4+20x^3)\cdot x-3\, \sqrt[3]{x^5+5x^4-5}\cdot 1}{x^2}=\\\\\\=\frac{(5x^5+20x^4)(x^5+5x^4-5)^{-\frac{2}{3}}-3\sqrt[3]{x^5+5x^4-5}}{x^2}=\\\\\\=\frac{5x^3+20x^2}{\sqrt[3]{(x^3+5x^4-5)^2}}-\frac{3\sqrt[3]{x^5+5x^4-5}}{x^2}$