Question

На решение есть ещё 40 минут

Answer 1

1.

$z = 9 {y}^{2} - 3 {x}^{4} + 2$

$z'_x = - 12 {x}^{3}$

$z'_y = 18y$

2.

$z = (7 {x}^{2} - 8 {x}^{2} {y}^{5} + 16y) {}^{2}$

$z'_x = 2(7 {x}^{2} - 8 {x}^{2} {y}^{5} + 16y) \times (14x - 14x {y}^{5} ) = \\$

$z'_y = 2(7 {x}^{2} - 8 {x}^{2} {y}^{5} + 16y) \times ( - 40 {x}^{2} {y}^{4} + 16) \\$

3.

$z = 10 \sin( {x}^{2} - 3 {y}^{5} )$

$z'_x = 10 \cos( {x}^{2} - 3 {y}^{5} ) \times 2x = \\ = 10x \cos( {x}^{2} - 3 {y}^{5} )$

$z'_y= 10 \cos( {x}^{2} - 3 {y}^{5} ) \times ( - 15 {y}^{4} ) = \\ = - 150 {y}^{4} \cos( {x}^{2} - 3 {y}^{5} )$

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4.

$z = 10 {x}^{2} y - 2x {y}^{2} + 6tgx$

$z'_x = 20xy - 2 {y}^{2} + \frac{6}{ \cos {}^{2} (x) } \\$

$z'_y = 10 {x}^{2} - 4xy$

$z''_{xx} = 20y + 6 \times ( - 2)( \cos(x) ) {}^{ - 3} \times ( - \sin(x)) = \\ = 20y + \frac{12 \sin(x) }{ \cos {}^{3} (x) }$

$z''_{yy} = - 4x$

$z''_{xy} = 20x - 4y$

5.

$z = 3 ln(10x - 6y)$

$z'_x = \frac{3}{10x - 6y} \times 10 = \frac{30}{10x - 6y} \\$

$z'_y = \frac{3}{10x - 6y} \times ( - 6) = - \frac{18}{10x - 6y} \\$

$z''_{xx} = 30 \times ( - (10x - 6y)) {}^{ - 2} \times 10 = \\ = - \frac{300}{ {(10x - 6y)}^{2} }$

$z''_{yy} = - 18 \times ( - (10x - 6y) {}^{ - 2} ) \times ( - 6) = \\ = - \frac{108}{ {(10x - 6y)}^{2} }$

$z''_{xy} = 30 \times ( - (10x - 6y) {}^{ - 2} ) \times ( - 6) = \\ = \frac{180}{ {(10x - 6y)}^{2} }$