Автор - YkharitonovaM

Найдите частные производные первого и второго порядка

Ответ

Проверено экспертом

Автор - NNNLLL54

z(x,y)=cos^3(-5x+3sqrt{y})\\\z'_{x}=3, cos^2(-5x+3sqrt{y})cdot (-sin(-5x+3sqrt{y}))cdot (-5)=\\=15cdot cos^2(-5x+3sqrt{y})cdot sin(-5x+3sqrt{y})=\\=frac{15}{2}cdot cos(-5x+3sqrt{y})cdot sin(-10x+6sqrt{y})\\\z'_{y}=3, cos^2(-5x+3sqrt{y})cdot (-sin(-5x+3sqrt{y}))cdot 3cdot frac{1}{2sqrt{y}}=\\=-frac{9}{2sqrt{y}}cdot cos^2(-5x+3sqrt{y})cdot sin(-5x+3sqrt{y})=\\=-frac{9}{4sqrt{y}}cdot cos(-5x+3sqrt{y})cdot sin(-10x+6sqrt{y})

z''_{xx}=frac{15}{2}cdot Big (-sin(-5x+3sqrt{y})cdot (-5)cdot sin(-10x+6sqrt{y})+\\+cos(-5x+3sqrt{y})cdot cos(-10x+6sqrt{y})cdot (-10)Big )=\\=frac{15}{2}cdot Big (5, sin(-5x+3sqrt{y})cdot sin(-10x+6sqrt{y})-\\-10, cos(-5x+3sqrt{y})cdot cos(-10x+6sqrt{y})Big )

z''_{xy}=frac{15}{2}cdot Big (-sin(-5x+3sqrt{y})cdot frac{3}{2sqrt{y}}cdot sin(-10x+6sqrt{y})+\\+cos(-5x+3sqrt{y})cdot cos(-10x+6sqrt{y})cdot frac{6}{2sqrt{y}}Big )

z''_{yy}=-frac{9}{4}cdot Big (-frac{1}{2sqrt{y^3}}cdot cos(-5x+3sqrt{y})cdot sin(-10x+6sqrt{y})+\\+frac{1}{sqrt{y}}cdot (-sin(-5x+3sqrt{y}))cdot frac{3}{2sqrt{y}}cdot sin(-10x+6sqrt{y})+\\+frac{1}{sqrt{y}}cdot cos(-5x+3sqrt{y})cdot cos(-10x+6sqrt{y})cdot frac{6}{2sqrt{y}}Big )

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