Question

Знайти усi похидни другого порядку функций z=x^2/1+2y

Answer 1

$z= \frac{x^2}{1+2y} \\\\z'_{x}= \frac{1}{1+2y} \cdot 2x\\\\z'_{y}=x^2\cdot \frac{-2}{(1+2y)^2} = -\frac{2x^2}{(1+2y)^2}\\\\z''_{xx}= \frac{1}{1+2y} \cdot 2\\\\z''_{xy}=2x\cdot \frac{-2}{(1+2y)^2} = -\frac{4x}{(1+2y)^2} =z''_{yx}\\\\z''_{yy}=-2x^2\cdot \frac{-2(1+2y)\cdot 2}{(1+2y)^4} = \frac{8x^2}{(1+2y)^3}$