Find the directional derivative of the function $f ( x , y , z ) = x e ^ { x y / z }$ at the point $P$ $( 3,0,1 )$ in the direction from $P$ toward the point $( 2,2,3 )$ ..
A) $\frac { 17 } { 3 }$
B) $6$
C) $\frac { 19 } { 3 }$
D) $\frac { 20 } { 3 }$
E)7
F) $\frac { 22 } { 3 }$
G) $\frac { 23 } { 3 }$
H)8

Question

Find the directional derivative of the function $f ( x , y , z ) = x e ^ { x y / z }$ at the point $P$ $( 3,0,1 )$ in the direction from $P$ toward the point $( 2,2,3 )$ ..
A) $\frac { 17 } { 3 }$ 
B) $6$ 
C) $\frac { 19 } { 3 }$
D) $\frac { 20 } { 3 }$
E)7
F) $\frac { 22 } { 3 }$  
G) $\frac { 23 } { 3 }$ 
H)8

Quizplus · Accepted Answer

The directional derivative is calculated using the gradient of $f$ and the unit vector in the direction from $P$ to $(2,2,3)$. The gradient at $P$ is $(e^0, 0, -3e^0)$. The direction vector is $(-1, 2, 2)$ with magnitude $\sqrt{9}$. The unit vector is $(-1/3, 2/3, 2/3)$. The dot product of the gradient and the unit vector gives $\frac{17}{3}$.

Find the Directional Derivative of the Function f(x,y,z)=xexy/zf ( x , y , z ) = x e ^ { x y / z }f(x,y,z)=xexy/z

Find the Directional Derivative of the Function $f ( x , y , z ) = x e ^ { x y / z }$