Question

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

• A. $\frac{4}{\sqrt{5}}$
• B. $-\frac{4}{\sqrt{5}}$
• C. $\frac{\sqrt{5}}{4}$
• D. $-\frac{\sqrt{5}}{4}$

Answer 1

The correct option is B $-\frac{4}{\sqrt{5}}$

$f(x,y,z)=2x^2+3y^2+z^2$
$▽f=4x\hat{i}+6y\hat{j}+2z\hat{k}$
$▽f{|}_{(2,1,3)}=8\hat{i}+6\hat{j}+6\hat{k}$
and $\overrightarrow{a}=\hat{i}-2\hat{k}$
Directional Derivative:
$=▽f.\hat{a}$
$=(8\hat{i}+6\hat{j}+6\hat{k}).\frac{(\hat{i}-2\hat{k})}{\sqrt{5}}$
$=\frac{8-12}{\sqrt{5}}=-\frac{4}{\sqrt{5}}$