Question

Найдите производные следующих функций:

Answer 1

1.

$y' = \frac{U'V - V'U}{ {V}^{2} } + F'G + G'F = \\ = \frac{(7 {x}^{6} - 10x)( {x}^{4} + 8) - 4 {x}^{3}( {x}^{7} - 5 {x}^{2} + 1) }{ {( {x}^{4} + 8) }^{2} } + 7 {x}^{6} {e}^{2x} + 2 { e}^{2x} \times {x}^{7} = \\ = \frac{7 {x}^{10} + 56 {x}^{6} - 10 {x}^{5} - 80x - 4 {x}^{10} + 20 {x}^{5} - 4 {x}^{3} }{ {( {x}^{4} + 8) }^{2} } + {x}^{6} {e}^{2x} (7 + 2x) = \\ = \frac{3 {x}^{10} + 56 {x}^{6} + 10 {x}^{5} - 4 {x}^{3} - 80x }{ {( {x}^{4} + 8)}^{2} } + {x}^{6} {e}^{2x} (2x + 7)$

2.

$y' = ( {(4 {x}^{5} + 3) }^{ - \frac{1}{6} } )' - ( - 2 \sin( \sqrt{7x} ) ) \times \sqrt{7} \times ( \sqrt{x} )' = \\ = - \frac{1}{6} {(4 {x}^{5} + 3)}^{ - \frac{7}{6} } \times 20 {x}^{4} + 2 \sqrt{7} \sin( \sqrt{7x} ) \times \frac{1}{2 \sqrt{x} } = \\ = - \frac{10 {x}^{4} }{3 \sqrt[6]{ {(4 {x}^{5} + 3)}^{7} } } + \frac{ \sqrt{7} \sin( \sqrt{7x} ) }{ \sqrt{x} }$