Question

Самостоятельная работа "Производная степенной функции". Решите, пожалуйста. Даю 35 баллов!!!

Answer 1

Ответ:

$1)y' = - \frac{1 }{2} \times 12 {x}^{11} = - 6 {x}^{11}$

$2)y' = 5 {x}^{4} - 3 {x}^{2}$

$3)y' = 3 {x}^{2} - {x}^{ - 2} = 3 {x}^{2} - \frac{1}{ {x}^{2} }$

$4)y' = 16 \times \frac{1}{2} {x}^{ - \frac{1}{2} } - 8x = \frac{8}{ \sqrt{x} } - 8x$

$5)y' = (5 {x}^{ - 1} + 2 {x}^{ - \frac{2}{3} } )' = 5 {x}^{ - 2} - \frac{4}{3} {x}^{ - \frac{5}{3} } = \frac{5}{ {x}^{2} } - \frac{4}{3x \sqrt[3]{ {x}^{2} } }$

$6)y' = (x + 7)' {x}^{2} + ( {x}^{2} )'(x + 7) = {x}^{2} + 2x(x + 7) = {x}^{2} + 2 {x}^{2} + 14x = 3 {x}^{2} + 14x$

$7)y' = \frac{1}{2} {(2x - 1)}^{ - \frac{1}{2} } \times (2x - 1)' \times ( {x}^{5} + 8) + 5 {x}^{4} \times \sqrt{2x - 1} = \\ = \frac{1}{2 \sqrt{2x - 1} } \times 2 \times ( {x}^{5} + 8) + 5 {x}^{4} \sqrt{2x - 1} = \\ = \frac{ {x}^{5} + 8}{ \sqrt{2x - 1} } + 5 {x}^{4} \sqrt{2x - 1}$

$8)y' = \frac{(2x + 3)'(2 - 3x) - (2 - 3x)'(2x + 3)}{ {(2 - 3x)}^{2} } = \frac{2(2 - 3x) + 3(2x + 3)}{ {(2 - 3x)}^{2} } = \frac{4 - 6x + 6x + 9}{ {(2 - 3x)}^{2} } = \frac{13}{ {(2 - 3x)}^{2} }$

$9)f'(x) = 5 {(x - 3)}^{4} \times (2x + 6) + 2 {(x - 3)}^{5} = {(x - 3)}^{4}(5(2x + 6) + 2(x - 3)) = {(x - 3)}^{4} (10x + 30 + 2x - 6) = {(x - 3)}^{4} (12x + 24)$

${(x - 3)}^{4} (12x + 24) = 0 \\ x1 = 3 \\ x2 = - 2$

$10)f'(x) = 2(x - 4) \sqrt{ x } + \frac{1}{2 \sqrt{x} } {(x - 4)}^{2} = (x - 4)(2 \sqrt{x} + \frac{x - 4}{2 \sqrt{x} } )$

$(x - 4)(2 \sqrt{x} + \frac{x - 4}{2 \sqrt{x} } ) = 0 \\ x1 = 4 \\ \\ 2 \sqrt{x} + \frac{x - 4}{2 \sqrt{x} } = 0 \\ 2 \sqrt{x} = - \frac{x - 4}{2 \sqrt{x} } \\ 4x = - (x - 4) \\ 4x = - x + 4 \\ 5x = 4 \\ x2 = \frac{4}{5}$

$11)f'(x) = 2 \times ( - \frac{1}{2}) {x}^{ - \frac{3}{2} } - 6x = - \frac{1}{x \sqrt{x} } - 6x$

$f'( \frac{1}{4} ) = - \frac{1}{ \frac{1}{4} \times \frac{1}{2} } - \frac{6}{4} = - 8 - 1.5 = - 9.5$

$12)f'(x) = 10x \sqrt[3]{x} + \frac{1}{3} {x}^{ - \frac{2}{3} } \times 5( {x}^{2} - 3) = 10x \sqrt[3]{x} + \frac{5( {x}^{2} - 3)}{3 \sqrt[3]{ {x}^{2} } }$

$f'(1) = 10 + \frac{ - 10}{3} = 10 - 3 \frac{1}{3} = 6 \frac{2}{3}$