Решебник по алгебре 11 класс Никольский Параграф 4. Производная Задание 57

\[\boxed{\mathbf{57}\mathbf{.}}\]

\[\textbf{а)}\ f(x) = \sin{2x};\ \ x \in R\]

\[f^{'}(x) = 2\cos{2x}.\]

\[\textbf{б)}\ f(x) = \cos{(3x + 1)};\ x \in R\]

\[f^{'}(x) = - \sin(3x + 1) \cdot 3 =\]

\[= - 3\sin(3x + 1).\]

\[\textbf{в)}\ f(x) = tg\ (2x - 3);\ \]

\[\ \ 2x - 3 \neq \frac{\pi}{2} + \pi k;\ \]

\[\ x \neq \frac{6 + \pi + 2\pi k}{4}\]

\[f^{'}(x) = \frac{1}{\cos^{2}(2x - 3)} \cdot 2 =\]

\[= \frac{2}{\cos^{2}{(2x - 3)}}.\]

\[\textbf{г)}\ f(x) = ctg\ ( - 5x);\]

\[\ - 5x \neq \pi k;\ \ x \neq - \frac{\text{πk}}{5}\]

\[f^{'}(x) = - \frac{- 5}{\sin^{2}{5x}} = \frac{5}{\sin^{2}{5x}}.\]