Решебник по алгебре и начала математического анализа 11 класс Алимов Задание 872

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\[1)\ f(x) = x^{2} \bullet \cos x\]

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

\[= \left( x^{2} \right)^{'} \bullet \cos x + x^{2} \bullet \left( \cos x \right)^{'} =\]

\[= 2x \bullet \cos x + x^{2} \bullet \left( - \sin x \right) =\]

\[= x \bullet \left( 2\cos x - x \bullet \sin x \right)\]

\[2)\ f(x) = x^{3} \bullet \ln x\]

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

\[= 3x^{2} \bullet \ln x + x^{3} \bullet \frac{1}{x} =\]

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

\[3)\ f(x) = 5x \bullet e^{x}\]

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

\[= 5e^{x} + 5x \bullet e^{x} =\]

\[= 5e^{x} \bullet (1 + x)\]

\[4)\ f(x) = x \bullet \sin{2x}\]

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

\[= (x)^{'} \bullet \sin{2x} + x \bullet \left( \sin{2x} \right)^{'} =\]

\[= 1 \bullet \sin{2x} + x \bullet 2\cos{2x} =\]

\[= \sin{2x} + 2x \bullet \cos{2x}\]

\[5)\ f(x) = e^{- x} \bullet \sin x\]

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

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

\[= - e^{- x} \bullet \sin x + e^{- x} \bullet \cos x =\]

\[= e^{x} \bullet \left( \cos x - \sin x \right)\]

\[6)\ f(x) = e^{x} \bullet \cos x\]

\[f^{'}(x) = \left( e^{x} \right)^{'} \bullet \cos x + e^{x} \bullet \left( \cos x \right)^{'} =\]

\[= e^{x} \bullet \cos x + e^{x} \bullet \left( - \sin x \right) =\]

\[= e^{x} \bullet \left( \cos x - \sin x \right).\]