Find the directional derivative to the given function at the specified point $P$ in the direction of the vector $\vec { u }$ $f ( x , y , z ) = 2 z \sqrt { x y } , P = ( 2,2,3 ) , \vec { u } = \langle 1,1,1 \rangle$
A) $10$

B) $10 \sqrt { 3 }$

C) $\frac { 10 } { 3 }$

D) $\frac { 10 } { \sqrt { 3 } }$

Question

Find the directional derivative to the given function at the specified point $P$ in the direction of the vector $\vec { u }$ $f ( x , y , z ) = 2 z \sqrt { x y } , P = ( 2,2,3 ) , \vec { u } = \langle 1,1,1 \rangle$
A) $10$

B) $10 \sqrt { 3 }$

C) $\frac { 10 } { 3 }$

D) $\frac { 10 } { \sqrt { 3 } }$

Quizplus · Accepted Answer

The directional derivative is found by taking the dot product of the gradient of the function at the point $P$ with the unit vector in the direction of $\vec{u}$. The gradient $\nabla f$ is $\left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)$. At $P = (2,2,3)$, $\nabla f = \left(3, 3, 2\sqrt{4}\right) = (3, 3, 4)$. The unit vector $\vec{u}$ is $\frac{1}{\sqrt{3}}\langle 1, 1, 1 \rangle$. The dot product is $(3, 3, 4) \cdot \frac{1}{\sqrt{3}}(1, 1, 1) = \frac{10}{\sqrt{3}}$.

Find the Directional Derivative to the Given Function at the Specified