Question

Найдите частные производные первого и второго порядка заданной функции: СРОЧНО!!!!!

Answer 1

Ответ:

$z = {e}^{3x - {y}^{3} }$

$z'_x = {e}^{3x - {y}^{3} } \times 3$

$z'_y = {e}^{3x - {y}^{3} } \times ( - 3 {y}^{2}) = - 3 {y}^{2} {e}^{3x - {y}^{3} }$

$z''_{xx} = 3 {e}^{3x - {y}^{3} } \times 3 = 9 {e}^{3x - {y}^{3} }$

$z''_{yy} = ( - 3 {y}^{2} ) {e}^{3x - {y}^{3} } + ( e {}^{3x - {y}^{3} } )'_y \times ( - 3 {y}^{2} ) = \\ = - 6y {e}^{ 3x - {y}^{3} } - 3 {y}^{2} {e}^{3x - {y}^{3} } \times ( - 3 {y}^{2} ) = \\ - 3y {e}^{3x - {y}^{3} } (2 + 3 {y}^{3} )$

$z''_{xy} = z''_{yx} = (3 {e}^{3x - {y}^{3} } )'_y = 3 {e}^{3x - {y}^{3} } \times ( - 3 {y}^{2} ) = \\ = - 9 {y}^{2} {e}^{3x - {y}^{3} }$