Решебник по алгебре 11 класс Никольский Параграф 4. Производная Задание 48

\[\boxed{\mathbf{48}\mathbf{.}}\]

\[\textbf{а)}\ f(x) = x^{12} + 12^{x};\ \ x \in R\]

\[f^{'}(x) = 12x^{11} + 12^{x}\ln 12.\]

\[\textbf{б)}\ f(x) = x^{20} - 3\sin x;\ x \in R\]

\[f^{'}(x) = 20x^{19} - 3\cos x\text{.\ }\]

\[\textbf{в)}\ f(x) = \ln x - \cos x;\ \ x > 0\]

\[f^{'}(x) = \frac{1}{x} + \sin x.\]

\[\textbf{г)}\ f(x) = \log_{4}x + x^{- 2};\ \ x > 0\]

\[f^{'}(x) = \frac{1}{x\ln 4} - 2x^{- 3}.\]

\[\textbf{д)}\ f(x) = x^{12} \cdot 12^{x};\ \ x \in R\]

\[f^{'}(x) = \left( x^{12} \right)^{'} \cdot 12^{x} +\]

\[+ x^{12} \cdot \left( 12^{x} \right) = 12x^{11} \cdot 12^{x} +\]

\[+ x^{12} \cdot 12^{x}\ln 12.\]

\[\textbf{е)}\ f(x) = x^{25} \cdot 4\cos x;\ \ x \in R\]

\[f^{'}(x) = \left( x^{25} \right)^{'} \cdot 4\cos x +\]

\[+ x^{25} \cdot \left( 4\cos x \right)^{'} =\]

\[= 25x^{24} \cdot 4\cos x +\]

\[+ x^{25} \cdot \left( - 4\sin x \right) = \ \]

\[= 100x^{24}\cos x - 4x^{25}\sin x.\]