Question

Помогите пожалуйста решить примеры с 468 по 474! Сложная тема :(

Answer 1

Ответ:

468

$y = \sqrt{2x} = \sqrt{2} \times \sqrt{x}$

$y '= \sqrt{2} \times \frac{1}{2} {x}^{ - \frac{1}{2} } = \frac{ \sqrt{2} }{2 \sqrt{x} } = \frac{1}{ \sqrt{2x} }$

$dy = \frac{dx}{ \sqrt{2x} }$

469

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

$y' = \frac{1}{2} {( {x}^{2} - 3x)}^{ - \frac{1}{2} } \times ( {x}^{2} - 3x) '= \\ = \frac{2x - 3}{2 \sqrt{ {x}^{2} - 3x } } \\ \\ dy= \frac{(2x - 3)dx}{2 \sqrt{ {x}^{2} - 3x } }$

470

$y = \frac{x}{x + 5}$

$y '= \frac{x'(x + 5) - (x + 5) '\times x}{ {(x + 5)}^{2} } = \frac{x + 5 - x}{ {(x + 5)}^{2} } = \frac{5}{ {(x + 5)}^{2} }$

$dy = \frac{5dx}{ {(x + 5)}^{2} }$

471

$y = \frac{x - 3}{x}$

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

$dy = \frac{3 {}^{} }{ {x}^{2} } dx$

472

$y = \frac{ {x}^{2} - 1}{x + 2}$

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

$dy = \frac{ {x}^{2} + 8x + 1}{ {(x + 4)}^{2} } dx$

473

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

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

$dy = \frac{ {x}^{2} - 1}{ {x}^{2} } dx$

474

$y = x(x - 3) = {x}^{2} - 3x$

$y' = 2x - 3 \\ dy = (2x - 3)dx$