Решебник по алгебре 11 класс Никольский Параграф 4. Производная Задание 30

\[\boxed{\mathbf{30}\mathbf{.}}\]

\[\textbf{а)}\ y = (x^{2} + 3x)(x - 1)\]

\[y^{'} = \left( x^{2} + 3x \right)^{'}(x - 1) +\]

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

\[= (2x + 3)(x - 1) +\]

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

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

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

\[\textbf{б)}\ y = (x^{2} - 8x)(x - 2)\]

\[y^{'} = \left( x^{2} - 8x \right)^{'}(x - 2) +\]

\[+ \left( x^{2} - 8x \right)(x - 2)^{'} =\]

\[= (2x - 8)(x - 2) +\]

\[+ \left( x^{2} - 8x \right) \cdot 1 = 2x^{2} - 8x -\]

\[- 4x + 16 + x^{2} - 8x =\]

\[= 3x^{2} - 20x + 16.\]

\[\textbf{в)}\ y = (5x^{2} - 3x + 2)(3x + 2)\]

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

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

\[= (10x - 3)(3x + 2) +\]

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

\[= 30x^{2} - 9x + 20x - 6 +\]

\[+ 15x^{2} - 9x + 6 = 45x^{2} + 2x.\]

\[\textbf{г)}\ y = (5x^{2} + 3x + 2)(3x - 2)\]

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

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

\[= (10x + 3)(3x - 2) +\]

\[+ \left( 5x^{2} + 3x + 2 \right) \cdot 3 =\]

\[= 30x^{2} + 9x - 20x - 6 +\]

\[+ 15x^{2} + 9x + 6 = 45x^{2} - 2x.\]

\[\textbf{д)}\ y = ( - x^{2} + 2)(3x^{2} + 2x)\]

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

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

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

\[+ \left( - x^{2} + 2 \right)(6x + 2) =\]

\[= - 6x^{3} - 4x^{2} - 6x^{3} + 12x -\]

\[- 2x^{2} + 4 = - 12x^{3} - 6x^{2} +\]

\[+ 12x + 4.\]

\[\textbf{е)}\ y = (4x^{2} + 6x - 1)(x^{2} - 3)\]

\[y^{'} = \left( 4x^{2} + 6x - 1 \right)^{'}\left( x^{2} - 3 \right) +\]

\[+ \left( 4x^{2} + 6x - 1 \right)\left( x^{2} - 3 \right)^{'} =\]

\[= (8x + 6) \cdot \left( x^{2} - 3 \right) +\]

\[+ \left( 4x^{2} + 6x - 1 \right) \cdot 2x =\]

\[= 8x^{3} + 6x^{2} - 24x - 18 +\]

\[+ 8x^{3} + 12x^{2} - 2x =\]

\[= 16x^{3} + 18x^{2} - 26x - 18.\]