Question

Найти производную функции y(x)=$\frac{a*x}{3}*\sqrt{b^2-\frac{a^2}{4}-\frac{x^2}{4} }$

Answer 1

$y(x)=\frac{ax}{3}\cdot \sqrt{b^2-\frac{a^2}{4}-\frac{x^2}{4}}\; ,\; \; a,b=const\\\\y'(x)=\frac{a}{3}\cdot \sqrt{b^2-\frac{a^2}{4}-\frac{x^2}{4}}+\frac{ax}{3}\cdot \frac{1}{2\sqrt{b^2-\frac{a^2}{4}-\frac{x^2}{4}}}\cdot (-\frac{2x}{4})=\\\\=\frac{a}{3}\cdot \sqrt{b^2-\frac{a^2}{4}-\frac{x^2}{4}}-\frac{ax^2}{6}\cdot \frac{1}{2\sqrt{\frac{4b^2-a^2-x^2}{4}}}=\\\\=\frac{a}{6}\cdot \sqrt{4b^2-a^2-x^2}-\frac{ax^2}{6}\cdot \frac{1}{\sqrt{4b^2-a^2-x^2}}$