Question

Сделайте математику пожалуйста, заранее спасибо

Answer 1

Ответ:

5.

$z = x \sin(xy) + y \cos(xy)$

$\frac{dz}{dx} = z'_x = (x)' \sin(xy) + ( \sin(xy))'_x \times x + y \times ( \cos(xy)) '_x = \\ = \sin(xy) + \cos(xy) \times y \times x - y \sin(xy) \times y = \\ = \sin( xy ) + xy \cos(xy) - {y}^{2} \sin(xy) = \\ = (1 - {y}^{2} ) \sin(xy) + xy \cos(xy)$

$\frac{ {d}^{2} z}{dx {}^{2} } =z''_{xx} = (1 - {y}^{2} ) \times ( \sin(xy))'_x + y \times (x' \cos(xy) + ( \cos(xy))'_x \times x) = \\ = (1 - {y}^{2} ) \times \cos(xy) \times y + y( \cos(xy) - \sin(xy) \times y \times x) = \\ = y(1 - {y}^{2} ) \cos(xy) + y \cos(xy) - x {y}^{2} \sin(xy) = \\ = (y - {y}^{3} + y) \cos(xy) - x {y}^{2} \sin(xy) = \\ = (2y - {y}^{3} ) \cos(xy) - x {y}^{2} \sin(xy)$