Question

Найти производную функции по направлению вектора

Answer 1

$z=-frac{x}{y^4},; M_0(-1,1),; overline {a}=overline {i}+overline {j}\\z'_{x}=-frac{1}{y^4},; z'_{x}(-1,1)=-1\\z'_{y}=frac{xcdot 4y^3}{y^8}=frac{4x}{y^5}; z'_{y}(-1,1)=-4\\overline {a}=(a_{x},a_{y})=(1,1),; |overline {a}|=sqrt{1^2+1^2}=sqrt2,\\overline {a^circ }=(frac{a_{x}}{|overline {a}|},frac{a_{y}}{|overline {a}|})=(frac{1}{sqrt2},frac{1}{sqrt2})$

$frac{partial z}{partial overline {a}}=z'_{x}(-1,1)cdot frac{a_{x}}{|overline{a}|}+z'_{y}(-1,1)cdot frac{a_{y}}{|overline {a}|}=-1cdot frac{1}{sqrt2}-4cdot frac{1}{sqrt2}=-frac{5}{sqrt2}$