Question

знайти похідні функції

Answer 1

1)

$y = 5 {x}^{3} - 3 {x}^{2} + 6 \\ y' = 5 \times 3 {x}^{3 - 1} - 3\times 2 {x}^{2 - 1} = \\ 15 {x}^{2} - 6x = 3x(5x - 2)$

2)

$y = \frac{2}{x} + \frac{x}{2} \\ y' = - \frac{2}{ {x}^{2} } + \frac{1}{2}$

3)

$y = (2x + 3)( {x}^{2} + 3x + 1) \\ y' = (2x + 3)'( {x}^{2} + 3x + 1) + (2x + 3)( {x}^{2} + 3x + 1)' = \\ 2( {x}^{2} + 3x + 1) + (2x + 3)(2x + 3) = \\ 2 {x}^{2} + 6x + 2 +4 {x }^{2} + 12x + 9 = \\ 6 {x}^{2} + 18x + 11$

4)

$y = \frac{ {x}^{3} }{4 - x} \\ y' = \frac{( {x}^{3}) '(4 - x) - (4 - x)' {x}^{3} }{(4 - x) {}^{2} } = \\ \frac{3 {x}^{2} (4 - x) - ( - {x}^{3} )}{(4 - x) {}^{2} } = \\ \frac{12 {x}^{2} - 3 {x}^{3} + {x}^{3} }{(4 - x) {}^{2} } = \frac{ 12 {x}^{2} - 2 {x}^{3} }{(4 - x) {}^{2} }$

5)

$y = \tg (2x + 3) \\ y' = \frac{2}{ \cos {}^{2} ( 2x + 3 ) }$