Question

Найти частные производные второго порядка функции.

Answer 1

1.

$z = \frac{1}{3} {x}^{3} - 5 {x}^{2} - 2xy - {y}^{2} + 20y \\$

$z'_x = {x}^{2} - 10x - 2y$

$z'_y = - 2x - 2y + 20$

$z''_{xx} = 2x - 10$

$z''_{yy} = - 2$

$z''_{xy} = ( {x}^{2} - 10x - 2y)'_y = - 2$

2.

$z = {2}^{( \frac{y}{x} + \frac{x}{y} )} \\$

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

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

$z''_{xx} = ln(2) \times ( ({2}^{( \frac{y}{x} + \frac{x}{y} ) } )'_x \times ( \frac{1}{y} - \frac{y}{ {x}^{2} } ) + ( \frac{1}{y} - \frac{y}{ {x}^{2} } )'_x \times {2}^{( \frac{y}{x} + \frac{x}{y} ) } ) = \\ = ln(2) \times ( ln(2) \times {2}^{( \frac{y}{x} + \frac{x}{y} ) } \times ( \frac{1}{y} - \frac{y}{ {x}^{2} } ) {}^{2} + 2y {x}^{ - 3} \times {2}^{( \frac{y}{x} + \frac{x}{y} ) } ) = \\ = ln(2) \times {2}^{( \frac{y}{x} + \frac{x}{y} ) } ( ln(2) \times {( \frac{1}{y} - \frac{y}{ {x}^{2} } ) }^{2} + \frac{2y}{ {x}^{3} } )$

$z''_{yy }= ln(2) \times (( {2}^{( \frac{y}{x} + \frac{x}{y} ) } )'_y \times ( \frac{1}{x} - \frac{x}{ {y}^{2} } ) + ( \frac{1}{x} - \frac{x}{ {y}^{2} } ) '_y \times {2}^{( \frac{y}{x} + \frac{x}{y} ) } ) = \\ = ln(2) \times ( ln(2) \times {2}^{( \frac{y}{x} + \frac{x}{y} ) } \times {( \frac{1}{x} - \frac{x}{ {y}^{2} } ) }^{2} + 2x {y}^{ - 3} \times {2}^{( \frac{y}{x} + \frac{x}{y} ) } ) = \\ = ln(2) \times {2}^{( \frac{y}{x} + \frac{x}{y} ) } ( ln(2) \times ( \frac{1}{x} - \frac{x}{ {y}^{2} } ) {}^{2} + \frac{2x}{ {y}^{3} } )$

$z''_{xy}= ln(2) \times (( {2}^{( \frac{y}{x} + \frac{x}{y} ) } )'_y \times ( \frac{1}{y} - \frac{y}{ {x}^{2} } ) + ( \frac{1}{y} - \frac{y}{ {x}^{2} } )'_y \times {2}^{( \frac{y}{x} + \frac{x}{y} ) } ) = \\ = ln(2) \times ( ln(2) \times {2}^{( \frac{y}{x} + \frac{x}{y} ) } \times ( \frac{1}{x} - \frac{x}{ {y}^{2} } )( \frac{1}{y} - \frac{y}{ {x}^{2} } ) + ( - {y}^{ - 2} - \frac{1}{ {x}^{2} } ) \times {2}^{( \frac{y}{x} + \frac{x}{y} ) } ) = \\ = ln(2) \times {2}^{( \frac{y}{x} + \frac{x}{y} ) } ( ln(2) \times ( \frac{1}{x} - \frac{x}{ {x}^{2} } ) ( \frac{1}{y} - \frac{y}{ {x}^{2} } ) - \frac{1}{ {y}^{2} } - \frac{1}{ {x}^{2} } )$