Question

Производная от (ln(1/x))^2

Answer 1

$y=(ln\frac{1}{x})^2\; \; ,\qquad (u^2)'=2u\cdot u'\; ,\; u=ln\frac{1}{x}\\\\y'=2\, ln\frac{1}{x}\cdot (ln\frac{1}{x})'=\Big [\; (lnu)'=\frac{1}{u}\cdot u'\; ,\; \; u=\frac{1}{x}\; \Big ]=\\\\=2\, ln\frac{1}{x}\cdot \frac{1}{1/x}\cdot (\frac{1}{x})'=2\, ln\frac{1}{x}\cdot x\cdot \frac{-1}{x^2}=-2\cdot \frac{1}{x}\cdot ln\frac{1}{x}$