Question

For a scalar function $f(x,y,z)=x^2+3y^2+2z^2$ the directional derivative at the point $P(1,2,-1)$ in the direction of a vector $\hat{i}-\hat{j}+2\hat{k}$ is

• A. $-18$
• B. $-3\sqrt{6}$
• C. $2\sqrt{6}$
• D. $18$

Answer 1

The correct option is B $-3\sqrt{6}$

Given,
$f(x,y,z)=x^2+3y^2+2z^2$
$gradf=\hat{i}\frac{\partial f}{\partial x}+\hat{j}\frac{\partial f}{\partial y}+\hat{k}\frac{\partial f}{\partial z}$
$▽f=\hat{i}\left(2x\right)+\hat{j}\left(6y\right)+\hat{k}\left(4z\right)$
$=2\hat{i}+12\hat{j}-4\hat{k};$ at $P(1,2,-1)$
The required diretional derivative $=(▽f).\hat{a}$
$=(▽f).\frac{(\hat{i}-\hat{j}+2\hat{k})}{\sqrt{1^2+(-1{)}^2+(2{)}^2}}$
$=\frac{(2\hat{i}+12\hat{j}-4\hat{k}).(\hat{i}-\hat{j}+2\hat{k})}{\sqrt{6}}$
$=\frac{2-12-8}{\sqrt{6}}=-3\sqrt{6}$