Question

f(x)=2/3(x+2)x-1; x-1 под корнем
Найдите производную

Answer 1

$F(x)= \frac{2}{3} (x+2) \sqrt{x-1}\\\\ F'(x)= (\frac{2}{3} (x+2))' \sqrt{x-1} + \frac{2}{3} (x+2) (\sqrt{x-1})'=$
$= \frac{2}{3} \sqrt{x-1}+\frac{1}{3} (x+2) \frac{1}{ \sqrt{x-1} } = \frac{2 \sqrt{x-1} }{3} + \frac{x+2}{3 \sqrt{x-1} } = \frac{2\sqrt{x-1}\cdot\sqrt{x-1}+x+2}{3 \sqrt{x-1} } =\\ = \frac{2\sqrt{(x-1)^2}+x+2}{3 \sqrt{x-1} } =\frac{2(x-1)+x+2}{3 \sqrt{x-1} } =\frac{2x-2+x+2}{3 \sqrt{x-1} } =\frac{3x}{3 \sqrt{x-1} } =\frac{x}{\sqrt{x-1} }$