Question

Find the directional derivative of f(x,y,z)=z^3−x^(2)y at the point (-3, -4, 1) in the direction of the vector v=⟨4,−3,1⟩.

Answer 1

$\nabla f=(-2xy,-x^2,3z^2).$ Hence at (-3,-4,1) the value is (-24,-9,3). Unit vector along the direction is $\frac{(4,-3,1)}{\sqrt{}4^2+(-3)^2+1^2}=(\frac{4}{\sqrt{26}},\frac{-3}{\sqrt{26}},\frac{1}{\sqrt{26}}).$ Hence required derivative= $-24.\frac{4}{\sqrt{26}}-9.\frac{-3}{\sqrt{26}}+3.\frac{1}{\sqrt{26}}=\frac{-66}{\sqrt{26}}$.