Question

Сделайте пожалуйста, заранее спасибо

Answer 1

1.

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

$dz = z'_x \times dx + z'_y \times dy$

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

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

$dz = - \frac{y}{ {(x + y)}^{2} } dx + \frac{x}{ {(x + y)}^{2} } dy = \\ = \frac{1}{ {(x + y)}^{2} } ( - ydx + xdy)$

2.

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

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

В точке М:

$z'_x(1; - 2) = - \frac{ (- 2)}{1} + 1 = 3$

$z'_y = \frac{1}{x}$

В точке М

$z'_y(1 ;- 2) = 1$