Функции многих переменных

[latex]z=\sin(\frac{x^3}{y^2}-8)\ \ \ \ \ \ \ \ \ \ \ \ M(2;\ 1) \\ \\ \\ z'_y=(\sin(\frac{x^3}{y^2}-8))'_z=\cos(\frac{x^3}{y^2}-8)*(\frac{x^3}{y^2}-8)'_y = \\ \\ \\ = \cos(\frac{x^3}{y^2}-8)*((\frac{x^3}{y^2})'_y-0)= \\ \\ \\ = \cos(\frac{x^3}{y^2}-8)*(\frac{0*y^2-x^3*2y}{y^4})= \cos(\frac{x^3}{y^2}-8)*(\frac{-2x^3y}{y^4})= \\ \\ \\ = -2\cos(\frac{x^3}{y^2}-8)*\frac{x^3}{y^3} \\ \\ \\ z'_y(M)=-2\cos(\frac{2^3}{1^2}-8)*\frac{2^3}{1^3}=-2\cos0*8=-2*1*8=-16[/latex]