Решебник по алгебре 11 класс Никольский Параграф 4. Производная Задание 72

\[\boxed{\mathbf{72}\mathbf{.}}\]

\[y = arcctg\ (x);\ \ x \in R\]

\[x = ctg\ y;\ \ y \in (0;\ \pi)\]

\[x^{'}(y) = - \frac{1}{\sin^{2}y};\]

\[f^{'}(x) = \frac{1}{- \frac{1}{\sin^{2}y}} =\]

\[= - \frac{1}{\frac{\sin^{2}y + \cos^{2}y}{\sin^{2}y}} =\]

\[= - \frac{1}{\frac{\sin^{2}y}{\sin^{2}y} + \frac{\cos^{2}y}{\sin^{2}y}} =\]

\[= - \frac{1}{1 + ctg^{2}y} = - \frac{1}{1 + x^{2}}.\]