Question

The directional derivative of the function f(x, y) = $x^2+y^2$ along a line directed from (0, 0) to (1, 1), evaluated at the point x = 1, y = 1 is

• A. 2
• B. $4\sqrt{2}$
• C. $2\sqrt{2}$
• D. $\sqrt{2}$

Answer 1

The correct option is C $2\sqrt{2}$

Directional derivative = $▽f.\frac{\overrightarrow{a}}{\left|\overrightarrow{a}\right|}$
$▽f$ = $\frac{\partial x^2}{\partial x}\hat{i}+\frac{\partial y^2}{\partial y}\hat{j}=2x\hat{i}+2y\hat{j}$
$\overrightarrow{a}$ = line joining (0, 0) and (1, 1)
=1$\hat{i}$ + $1\hat{j}$
$\overrightarrow{a}$ = $\hat{i}$+$\hat{j}$
Directional derivative = $\frac{(2x\hat{i}+2y\hat{j})(\hat{i}+\hat{j})}{\sqrt{2}}$
=$\frac{2x+2y}{\sqrt{2}}$
Directional derivative at (1, 1) = $\frac{2+2}{\sqrt{2}}$
=$2\sqrt{2}$