[tex]f(x,y,z)=x+\dfrac yz[/tex]
[tex]\implies \nabla f(x,y,z)=\left(1,\dfrac1z,-\dfrac y{z^2}\right)[/tex]
[tex]\implies \nabla f(2,-3,5)=\left(1,\dfrac15,\dfrac3{25}\right)[/tex]
The maximum rate of change is then
[tex]\|\nabla f(2,-3,5)\|=\dfrac{\sqrt{659}}{25}[/tex]
which occurs in the direction of [tex]\nabla f(2,-3,5)[/tex] above.
