Решебник по алгебре 11 класс Никольский Параграф 4. Производная Задание 45

\[\boxed{\mathbf{45}\mathbf{.}}\]

\[\textbf{а)}\ f(x) = \log_{2}x;\ \ x > 0\]

\[f^{'}(x) = \left( \log_{2}x \right)^{'} = \frac{1}{x\ln 2}.\]

\[\textbf{б)}\ f(x) = \lg x;\ \ x > 0\]

\[f^{'}(x) = \left( \lg x \right)^{'} = \left( \log_{10}x \right)^{'} =\]

\[= \frac{1}{x\ln 10}.\]

\[\textbf{в)}\ f(x) = 4\log_{2}x + 3\ln x -\]

\[- 2\lg x;\ \ x > 0\]

\[f^{'}(x) = 4 \cdot \left( \log_{2}x \right)^{'} +\]

\[+ 3 \cdot \left( \ln x \right)^{'} - 2 \cdot \left( \lg x \right)^{'} =\]

\[= 4 \cdot \frac{1}{x\ln 2} + 3 \cdot \frac{1}{x} - 2 \cdot \frac{1}{x\ln 10} =\]

\[= \frac{4}{x\ln 2} + \frac{3}{x} - \frac{2}{x\ln 10}.\]

\[\textbf{г)}\ f(x) = 5\log_{3}x - 6\ln x +\]

\[+ 7\lg x;\ \ x > 0\ \]

\[f^{'}(x) = 5 \cdot \left( \log_{3}x \right)^{'} -\]

\[- 6 \cdot \left( \ln x \right)^{'} + 7 \cdot \left( \lg x \right)^{'} =\]

\[= 5 \cdot \frac{1}{x\ln 2} - 6 \cdot \frac{1}{x} + 7 \cdot \frac{1}{x\ln 10} =\]

\[= \frac{5}{x\ln 2} - \ \frac{6}{x} + \frac{7}{x\ln 10}.\]