Question

Помогите, пожалуйста с этим заданием! ДАЮ 30 БАЛЛОВ

$\frac{dy}{dx}$ и $\frac{d^2y}{dx^2}$ : a) $y=xe^{x^2}$

Answer 1

Ответ:

$y = x {e}^{ {x}^{2} }$

$\frac{dy}{dx} = x' {e}^{ {x}^{2} } + (e {}^{ {x}^{2} } )' \times x = \\ = {e}^{ {x}^{2} } + {e}^{ {x}^{2} } \times 2x \times x = \\ = {e}^{ {x}^{2} } (1 + 2 {x}^{2} )$

$\frac{ {d}^{2} y}{dx {}^{2} } = ( {e}^{ {x}^{2} } )' \times ( 1+ 2 {x}^{2}) + (1 + 2 {x}^{2}) 'e {}^{ {x}^{2} } = \\ = {e}^{ {x}^{2} } \times 2x \times (1 + 2 {x}^{2} ) + 4x \times {e}^{2 {x}^{2} } = \\ = {e}^{ {x}^{2} } (2x + 4 {x}^{3} + 4x) = {e}^{ {x}^{2} } (4 {x}^{3} + 6x)$