Решебник по алгебре и начала математического анализа 11 класс Колягин Задание 176

\[1)\ f(y) = \sqrt{y^{2} - 1};\]

\[y = g(x) = \sqrt{x^{2} + 1};\]

\[f\left( g(x) \right) = \sqrt{\left( x^{2} + 1 \right) - 1} =\]

\[= \sqrt{x^{2}} = |x|.\]

\[x < 0:\]

\[y = - x;\ \ \ \]

\[y^{'} = - 1.\]

\[x > 0:\]

\[y = x;\]

\[y^{'} = 1.\]

\[f^{'}\left( g(x) \right) = \left\{ \begin{matrix} - 1\ при\ x < 0 \\ 1\ \ \ \ при\ x > 1 \\ \end{matrix} \right.\ .\]

\[2)\ f(y) = \sqrt{1 - y^{2}};\]

\[y = g(x) = \cos x;\]

\[f\left( g(x) \right) = \sqrt{1 - \cos^{2}x} =\]

\[= \sqrt{\sin^{2}x} = \left| \sin x \right|.\]

\[- \pi + 2\pi n < x < 2\pi n:\]

\[y = - \sin x;\]

\[y^{'} = - \cos x.\]

\[2\pi n < x < \pi + 2\pi n:\]

\[y = \sin x;\]

\[y^{'} = \cos x.\]