Question

Find the maximum rate of change of f(x,y,z=x y/z at the point (2, -3, 5 and the direction in which it occurs. maximum rate of change:

Answer 1

$f(x,y,z)=x+\dfrac yz$
$\implies \nabla f(x,y,z)=\left(1,\dfrac1z,-\dfrac y{z^2}\right)$
$\implies \nabla f(2,-3,5)=\left(1,\dfrac15,\dfrac3{25}\right)$

The maximum rate of change is then

$\|\nabla f(2,-3,5)\|=\dfrac{\sqrt{659}}{25}$

which occurs in the direction of $\nabla f(2,-3,5)$ above.