Find the directional derivative of the function $f ( x , y , z ) = \sqrt { x y z }$ at the point $( 2,4,2 )$ in the direction of the vector $\{ 4,2 , - 4 \}$ .
A) $- \frac { 1 } { 6 }$

B) $- \frac { 1 } { 4 }$
C) $- \frac { 1 } { 2 }$
D)0
E) $\frac { 1 } { 2 }$
F) $\frac { 1 } { 4 }$

G) $\frac { 1 } { 6 }$
H) $\frac { 1 } { 8 }$

Question

Find the directional derivative of the function $f ( x , y , z ) = \sqrt { x y z }$ at the point $( 2,4,2 )$ in the direction of the vector $\{ 4,2 , - 4 \}$ .
A) $- \frac { 1 } { 6 }$

B) $- \frac { 1 } { 4 }$
C) $- \frac { 1 } { 2 }$ 
D)0
E) $\frac { 1 } { 2 }$ 
F) $\frac { 1 } { 4 }$

G) $\frac { 1 } { 6 }$ 
H) $\frac { 1 } { 8 }$

Quizplus · Accepted Answer

The directional derivative of a function $f(x, y, z)$ at a point $(x_0, y_0, z_0)$ in the direction of a vector $\mathbf{v} = \{a, b, c\}$ is given by $\nabla f \cdot \mathbf{u}$, where $\nabla f$ is the gradient of $f$ and $\mathbf{u}$ is the unit vector in the direction of $\mathbf{v}$. First, we find the gradient of $f(x, y, z) = \sqrt{xyz}$:$$\nabla f = \left( \frac{1}{2} \sqrt{\frac{yz}{x}}, \frac{1}{2} \sqrt{\frac{xz}{y}}, \frac{1}{2} \sqrt{\frac{xy}{z}} \right)$$At the point $(2, 4, 2)$, this becomes:$$\nabla f(2, 4, 2) = \left( \frac{1}{2} \sqrt{\frac{4 \cdot 2}{2}}, \frac{1}{2} \sqrt{\frac{2 \cdot 2}{4}}, \frac{1}{2} \sqrt{\frac{2 \cdot 4}{2}} \right) = \left( 1, \frac{1}{2}, 1 \right)$$The given direction vector is $\mathbf{v} = \{4, 2, -4\}$. We normalize it to get the unit vector $\mathbf{u}$:$$\mathbf{u} = \frac{\mathbf{v}}{\|\mathbf{v}\|} = \frac{\{4, 2, -4\}}{\sqrt{4^2 + 2^2 + (-4)^2}} = \frac{\{4, 2, -4\}}{\sqrt{16 + 4 + 16}} = \frac{\{4, 2, -4\}}{\sqrt{36}} = \frac{\{4, 2, -4\}}{6}$$The directional derivative is then:$$\nabla f(2, 4, 2) \cdot \mathbf{u} = \left( 1, \frac{1}{2}, 1 \right) \cdot \left( \frac{4}{6}, \frac{2}{6}, \frac{-4}{6} \right) = \frac{4}{6} + \frac{1}{6} - \frac{4}{6} = \frac{1}{6}$$

Find the Directional Derivative of the Function f(x,y,z)=xyzf ( x , y , z ) = \sqrt { x y z }f(x,y,z)=xyz​

Find the Directional Derivative of the Function $f ( x , y , z ) = \sqrt { x y z }$