Question

Решите эти примеры , тема Частные производные ,

Answer 1

1.

$u = \frac{1}{ \sqrt[3]{x}} - \frac{1}{ \sqrt[3]{t} } = {x}^{ - \frac{1}{3} } - {t}^{ - \frac{1}{3} } \\$

$u'_x = - \frac{1}{3} {x}^{ - \frac{4}{3} } = - \frac{1}{3x \sqrt[3]{x} } \\$

$u'_t = - ( - \frac{1}{ 3} ) {t}^{ - \frac{4}{3} } = \frac{1}{3t \sqrt[3]{t} } \\$

2.

$c = \sqrt{ {a}^{2} + {b}^{2} - 2ab \cos( \alpha ) } = ( {a}^{2} + {b}^{2} - 2ab \cos( \alpha ) ) {}^{ \frac{1}{2} } \\$

$c'_a = \frac{1}{2} {( {a}^{2} + b {}^{2} - 2ab \cos( \alpha )) }^{ - \frac{1}{2} } \times (2a - 2b \cos( \alpha )) = \\ = \frac{2a - 2b \cos( \alpha ) }{2 \sqrt{ {a}^{2} + {b}^{2} - 2ab \cos( \alpha ) } } = \frac{a - b \cos( \alpha ) }{ \sqrt{ {a}^{2} + {b}^{2} - 2ab \cos( \alpha ) } }$

$c'_b = \frac{1}{2 \sqrt{ {a}^{2} + {b}^{2} - 2ab \cos( \alpha ) } } \times (2b - 2a \cos( \alpha )) = \\ = \frac{b - a \cos( \alpha ) }{ \sqrt{ {a}^{2} + {b}^{2} - 2ab \cos( \alpha ) } }$