Question

Найти полный дифференциал

Answer 1

$4)\; \; u=\frac{z}{x^2}\cdot e^{y}\\\\du=u'_{x}\, dx+u'_{y}\, dy+u'_{x}\, dz\\\\du=-\frac{2z\cdot e^{y}}{x^3}\, dx+\frac{z}{x^2}\cdot e^{y}\, dy+\frac{e^{y}}{x^2}\, dz$

$5)\; \; u=\frac{3}{2}\cdot sin(\frac{2}{3}x-y)\\\\du=u'_{x}\, dx+u'_{y}\, dy\\\\du=\frac{3}{2}\cdot cos(\frac{2}{3}x-y)\cdot \frac{2}{3}\, dx+\frac{3}{2}\cdot cos(\frac{2}{3}x-y)\cdot (-1)\, dy=\\\\=\frac{3}{2}\cdot cos(\frac{2}{3}x-y)\cdot (\frac{2}{3}\, dx-dy)$

$9)\; \; f(x,y,z)=\frac{x+2y}{z-y}\\\\d(x,y,z)=\frac{1}{z-y}\cdot 1\cdot dx+\frac{2(z-y)+(x+2y)}{(z-y)^2}\cdot dy-\frac{(x+2y)\cdot 1}{(z-y)^2}\cdot dz\\\\d(1,-2,3)=\frac{1}{3+2}\cdot dx+\frac{2(1+2)+(1-4)}{(3+2)^2}\cdot dy-\frac{1-4}{(3+2)^2}\cdot dz=\\\\=\frac{1}{5}\cdot dx+\frac{3}{25}\cdot dy+\frac{3}{25}\cdot dz$