u`_(x)=(xz^4/y)`_(x) + (xzy^5)`_(x) + (y/z^2)`_(x)=
=(z^4/y)*(x`)+(zy^5)*(x`)+0=(z^4/y)+(zy^5)
u`_(y)=(xz^4/y)`_(y) + (xzy^5)`_(y) + (y/z^2)`_(y)=
=(xz^4)*(1/y)`+xz*(y^5)`+(1/z^2)y`=
=(xz^4)*(-1/y^2)+5xz*y^4+(1/z^2)
u`_(z)=(xz^4/y)`_(z) + (xzy^5)`_(z) + (y/z^2)`_(z)=
=(x/y)*(z^4)`+xy^5*(z)`+y*(z^(-2))`=
=(4xz^3/y)+xy^5-2yz^(-3)
M(1;1;-1)
u`_(x)(M)=1-1=0
u`_(y)(M)=-1-5+1=-5
u`_(z)(M)=-4+1+2=-1
vector{MP}=(3-1;-5-1;2-(-1))=(2;-6;3)
|vector{MP}|=sqrt(2^2+(-6)^2+3^2)=sqrt(49)=7
Направляющие косинусы вектора vector{MP}
cos α =2/7
cos β =-6/7
cos γ =3/7
О т в е т.
u`_(MP)(M)=u`_(x)(M)cos α +u`_(y)(M)cos β +u`_(z)(M)cos γ =
=0*(2/7)-5*(-6/7)+1*(3/7)=33/7