What is the maximum rate of change of $f(x,y,z) = x + \frac yz$ at the point $(5, -3, 5)$ and the direction in which it occurs?
As an answer I got $1.02683981223947$ for the maximum rate of change. Being the magnitude of the gradient. However for the unit vector I got $\left \langle 1, \frac15, \frac3{25}\right \rangle$. I got that from $dx=1,\,dy=\frac1z,\,dz=-\frac y{z^2}$. The maximum rate of change I got was correct but the unit vector was incorrect (In all three parts)
Best Answer
$( 1,\frac15,\frac 3 {25}) $ isn't a unit vector; divide by its magnitude (the value you mentioned above )