Решебник по алгебре 11 класс Никольский Параграф 4. Производная Задание 18

\[\boxed{\mathbf{18}\mathbf{.}}\]

\[\textbf{а)}\ y = x^{3} + x^{2} + x;\ \ x \in R\]

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

\[= 3x^{2} + 2x + 1.\ \]

\[\textbf{б)}\ y = x^{3} - x^{2} - x;\ \ x \in R\]

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

\[= 3x^{2} - 2x - 1.\]

\[\textbf{в)}\ y = 5x^{3};\ \ x \in R\]

\[y^{'} = 5 \cdot \left( x^{3} \right)^{'} = 5 \cdot 3x^{2} = 15x^{2}.\]

\[\textbf{г)}\ y = - x^{3};\ \ x \in R\]

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

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

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

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

\[= 2 \cdot 3x^{2} - 3 \cdot 2x + 1 = 6x^{2} -\]

\[- 6x + 1.\]

\[\textbf{е)}\ y = 3x^{3} - 4x + 2;\ x \in R\]

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

\[= 3 \cdot 3x^{2} - 4 \cdot 1 + 0 = 9x^{2} - 4.\]

\[\textbf{ж)}\ y = - x^{3} + 5x^{2} - 8x + 13;\]

\[\ \ \ x \in R\]

\[y^{'} = - 1 \cdot \left( x^{3} \right)^{'} + 5 \cdot \left( x^{2} \right)^{'} -\]

\[- 8x^{'} + 13^{'} = - 1 \cdot 3x^{2} +\]

\[+ 5 \cdot 2x - 8 \cdot 1 + 0 =\]

\[= - 3x^{2} + 10x - 8.\]

\[\textbf{з)}\ y = ax^{3} + bx^{2} + cx + d;\ \]

\[\ x \in R\]

\[y^{'} = a \cdot \left( x^{3} \right)^{'} + b \cdot \left( x^{2} \right)^{'} + cx^{'} +\]

\[+ d^{'} = a \cdot 3x^{2} + b \cdot 2x +\]

\[+ c \cdot 1 + 0 =\]

\[= 3ax^{2} + 2bx + c.\]