Find the directional derivative of $f ( x , y ) = 2 \sqrt { x } - y ^ { 3 }$ at the point (1, 3) in the direction toward the point (3, 1). Select the correct answer.
A) $7 \sqrt { 2 }$
B) $14 \sqrt { 2 }$
C) $\sqrt { 2 }$
D) 28
E) none of these

Question

Quizplus · Accepted Answer

The directional derivative of a function $f(x,y)$ in the direction of a unit vector $	extbf{u}$ is given by $
abla f \cdot 	extbf{u}$.

First, we need to find the unit vector in the direction of $(3,1)$:
$$	extbf{u} = \frac{(3-1, 1-3)}{\sqrt{(3-1)^2+(1-3)^2}} = \frac{(2,-2)}{2\sqrt{2}} = \frac{1}{\sqrt{2}}(1,-1)$$

Next, we need to find the gradient of $f(x,y)$:
$$
abla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}ight) = \left(\frac{1}{\sqrt{x}}, -3y^2ight)$$

Evaluate the gradient at the point $(1,3)$:
$$
abla f(1,3) = \left(\frac{1}{\sqrt{1}}, -3(3^2)ight) = (1,-27)$$

Now we can find the directional derivative:
$$
abla f(1,3) \cdot 	extbf{u} = (1,-27) \cdot \frac{1}{\sqrt{2}}(1,-1) = \frac{1}{\sqrt{2}}(1 \cdot 1 - 27 \cdot 1) = -\frac{26}{\sqrt{2}} = -13\sqrt{2}$$

Therefore, the correct answer is B.

Find the Directional Derivative Of f(x,y)=2x−y3f ( x , y ) = 2 \sqrt { x } - y ^ { 3 }f(x,y)=2x​−y3

Find the Directional Derivative Of $f ( x , y ) = 2 \sqrt { x } - y ^ { 3 }$