Решебник по алгебре и начала математического анализа 10 класс Алимов Задание 870

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\[1)\ f(x) = e^{x} - \sin x\]

\[f^{'}(x) = \left( e^{x} \right)^{'} - \left( \sin x \right)^{'} =\]

\[= e^{x} - \cos x\]

\[2)\ f(x) = \cos x - \ln x\]

\[f^{'}(x) = \left( \cos x \right)^{'} - \left( \ln x \right)^{'} =\]

\[= - \sin x - \frac{1}{x}\]

\[3)\ f(x) = \sin x - \sqrt[3]{x}\]

\[f^{'}(x) = \left( \sin x \right)^{'} - \left( x^{\frac{1}{3}} \right)^{'} =\]

\[= \cos x - \frac{1}{3} \bullet x^{- \frac{2}{3}} =\]

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

\[4)\ f(x) = 6x^{4} - 9e^{x}\]

\[f^{'}(x) = 6 \bullet \left( x^{4} \right)^{'} - 9 \bullet \left( e^{x} \right)^{'} =\]

\[= 6 \bullet 4x^{3} - 9e^{x} = 24x^{3} - 9e^{x}\]

\[5)\ f(x) = \frac{5}{x} + 4e^{x}\]

\[f^{'}(x) = 5 \bullet \left( \frac{1}{x} \right)^{'} + 4 \bullet \left( e^{x} \right)^{'} =\]

\[= 5 \bullet \left( - \frac{1}{x^{2}} \right) + 4e^{x} = - \frac{5}{x^{2}} + 4e^{x}\]

\[6)\ f(x) = \frac{1}{3x^{3}} + \frac{1}{2}\ln x\]

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

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

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