Question

Найти частные производные первого порядка
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Answer 1

3.

$g(x,y) = \frac{ \sqrt{x} }{ {y}^{2} } = \sqrt{x} {y}^{ - 2} \\$

$g'_x = \frac{1}{2} {x}^{ - \frac{1}{2} } {y}^{ - 2} = \frac{1}{2 \sqrt{x} {y}^{2} } \\$

$g'_y = \sqrt{x} \times ( - 2 {y}^{ - 3} ) = - \frac{2 \sqrt{x} }{ {y}^{3} } \\$

6.

$F(x,y) = arcsin( {x}^{2} - 4y)$

$F'_x = \frac{1}{ \sqrt{1 - {( {x}^{2} - 4y) }^{2} } } \times 2x = \frac{2x}{ \sqrt{1 -( {x}^{4} - 8y {x}^{2} + 16 {y}^{2}) } } = \\ = \frac{2x}{ \sqrt{1 - {x}^{4} + 8x {}^{2} {y}^{} - 16 {y}^{2} } }$

$F'_y = \frac{1}{ \sqrt{1 - {( {x}^{2} - 4y)}^{2} } } \times ( - 4) = \\ = - \frac{4}{ \sqrt{1 - {x}^{4} + 8 {x}^{2}y {}^{} - 16 {y}^{2} } }$