Question

A derivada direcional da função f(x,y,z)=xyz no ponto P(1;3;3) na direção do vetor v=i+2j+2k vale:
( ) 1
( ) 9
( ) 1/3
( ) 7
( ) 3
( ) -1

Answer 1

$f(x,y,z)=xyz\\\\ \frac{ \partial f}{ \partial x} = yz\\\\\frac{ \partial f}{ \partial y}=xz\\\\\frac{ \partial f}{ \partial z}=xy \\\\ \boxed{\nabla f = (xy ,xz,xy)}\\\\\\ \vec V=(1,2,2)\\\\|\vec V| = \sqrt{1^2+2^2+2^2} =3\\\\\\\\ D_u f =\nabla . \frac{\vec V}{|\vec V|} \\\\ D_u f = (yz,xz,xy) \frac{(1,2,2)}{3} \\\\ D_u f = \frac{yz+2xz+2xy}{3}\\\\ D_u f(1,3,3) = \frac{3*3+2*1*3+21*3}{3}=7$