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Find-the-directional-derivatives-of-the-function-f-x-y-z-2x-2-3y-2-z-2-at-the-point-p-2-1-3-




Question Number 187255 by Spillover last updated on 15/Feb/23
Find the directional derivatives of the  function   f(x,y,z)=2x^2 +3y^2 +z^2  at the point p(2,1,3)
$${Find}\:{the}\:{directional}\:{derivatives}\:{of}\:{the} \\ $$$${function}\: \\ $$$${f}\left({x},\mathrm{y},\mathrm{z}\right)=\mathrm{2}{x}^{\mathrm{2}} +\mathrm{3}{y}^{\mathrm{2}} +{z}^{\mathrm{2}} \:{at}\:{the}\:{point}\:{p}\left(\mathrm{2},\mathrm{1},\mathrm{3}\right) \\ $$
Answered by MikeH last updated on 15/Feb/23
D_p (x,y,z) = (p_x ,p_y ,p_z )•(f_x ,f_y ,f_z )  ⇒ D_p (x,y,z) = 2(4x)+1(6y)+3(2z)  ⇒ D_p (x,y,z) = 8x + 6y + 6z
$${D}_{{p}} \left({x},{y},{z}\right)\:=\:\left({p}_{{x}} ,{p}_{{y}} ,{p}_{{z}} \right)\bullet\left({f}_{{x}} ,{f}_{{y}} ,{f}_{{z}} \right) \\ $$$$\Rightarrow\:{D}_{{p}} \left({x},{y},{z}\right)\:=\:\mathrm{2}\left(\mathrm{4}{x}\right)+\mathrm{1}\left(\mathrm{6}{y}\right)+\mathrm{3}\left(\mathrm{2}{z}\right) \\ $$$$\Rightarrow\:{D}_{{p}} \left({x},{y},{z}\right)\:=\:\mathrm{8}{x}\:+\:\mathrm{6}{y}\:+\:\mathrm{6}{z} \\ $$

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