Решебник по алгебре и начала математического анализа 11 класс Алимов Задание 953

\[\boxed{\mathbf{953}\mathbf{.}}\]

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

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

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

\[f^{'}(x) = 2x \bullet \cos x - x^{2} \bullet \sin x.\]

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

\[= \left( 2 - x^{2} \right) \bullet \cos x - 4x \bullet \sin x.\]

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

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

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

\[= 3x^{2} \bullet \sin x + x^{3} \bullet \cos x.\]

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

\[= \left( 6x - x^{3} \right) \bullet \sin x + 6x^{2} \bullet \cos x.\]

\[3)\ f(x) = x^{5} + 2x^{3} - x^{2} + 2\]

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

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

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

\[= 5x^{4} + 2 \bullet 3x^{2} - 2x + 0 =\]

\[= 5x^{4} + 6x^{2} - 2x.\]

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

\[= 5 \bullet \left( x^{4} \right)^{'} + 6 \bullet \left( x^{2} \right)^{'} - (2x)^{'};\]

\[f^{''}(x) = 5 \bullet 4x^{3} + 6 \bullet 2x - 2 =\]

\[= 20x^{3} + 12x - 2.\]

\[4)\ f(x) = x^{4} - 3x^{3} + 5x + 6\]

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

\[= \left( x^{4} \right)^{'} - 3 \bullet \left( x^{3} \right)^{'} + (5x + 6)^{'};\]

\[f^{'}(x) = 4x^{3} - 3 \bullet 3x^{2} + 5 =\]

\[= 4x^{3} - 9x^{2} + 5.\]

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

\[= 4 \bullet \left( x^{3} \right)^{'} - 9 \bullet \left( x^{2} \right)^{'} + (5)^{'};\]

\[f^{''}(x) = 4 \bullet 3x^{2} - 9 \bullet 2x + 0 =\]

\[= 12x^{2} - 18x.\]