Решебник по алгебре и начала математического анализа 10 класс Алимов Задание 843

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\[1)\ f(x) = \sqrt{\frac{2x - 1}{3}} + \ln\frac{2x + 3}{5}\]

\[f^{'}(x) = {\left( \frac{2}{3}x - \frac{1}{3} \right)^{\frac{1}{2}}}^{'} + \left( \ln\left( \frac{2}{5}x + \frac{3}{5} \right) \right)^{'} =\]

\[= \frac{1}{2} \bullet \frac{2}{3} \bullet \left( \frac{2}{3}x - \frac{1}{3} \right)^{- \frac{1}{2}} + \frac{\frac{2}{5}}{\frac{2}{5}x + \frac{3}{5}} =\]

\[= \frac{1}{3\sqrt{\frac{2}{3}x - \frac{1}{3}}} + \frac{2}{2x + 3} =\]

\[= \frac{1}{\sqrt{6x - 3}} + \frac{2}{2x + 3}.\]

\[2)\ f(x) = \sqrt{\frac{1 - x}{6}} - 2\ln\frac{2 - 5x}{3}\]

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

\[= {\left( \frac{1}{6} - \frac{1}{6}x \right)^{\frac{1}{2}}}^{'} - 2 \bullet \left( \ln\left( \frac{2}{3} - \frac{5}{3}x \right) \right)^{'} =\]

\[= \frac{1}{2} \bullet \left( - \frac{1}{6} \right) \bullet \left( \frac{1}{6} - \frac{1}{6}x \right)^{- \frac{1}{2}} - 2 \bullet \frac{- \frac{5}{3}}{\frac{2}{3} - \frac{5}{3}x} =\]

\[= - \frac{1}{12\sqrt{\frac{1}{6} - \frac{1}{6}x}} + \frac{2 \bullet 5}{2 - 5x} =\]

\[= \frac{10}{2 - 5x} - \frac{1}{2\sqrt{6 - 6x}}.\]

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

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

\[= 2 \bullet \left( e^{\frac{1}{3} - \frac{1}{3}x} \right)^{'} + 3 \bullet \left( \cos\left( \frac{1}{2} - \frac{1}{2}x \right) \right)^{'} =\]

\[= 2 \bullet \left( - \frac{1}{3} \right) \bullet e^{\frac{1}{3} - \frac{1}{3}x} + 3 \bullet \left( - \frac{1}{2} \right) \bullet \left( - \sin{\frac{1}{2} - \frac{1}{2}x} \right) =\]

\[= \frac{3}{2}\sin\frac{1 - x}{2} - \frac{2}{3}e^{\frac{1 - x}{3}}.\text{\ \ }\]

\[4)\ f(x) = 3e^{\frac{2 - x}{3}} - 2\sin\frac{1 + x}{4}\]

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

\[= 3 \bullet \left( e^{\frac{2}{3} - \frac{1}{3}x} \right)^{'} - 2 \bullet \left( \sin\left( \frac{1}{4} + \frac{1}{4}x \right) \right)^{'} =\]

\[= 3 \bullet \left( - \frac{1}{3} \right) \bullet e^{\frac{2}{3} - \frac{1}{3}x} - 2 \bullet \frac{1}{4} \bullet \cos\left( \frac{1}{4} + \frac{1}{4}x \right) =\]

\[= - e^{\frac{2 - x}{3}} - \frac{1}{2}\cos\frac{1 + x}{4}.\ \]