Answer
$1, \lt 0,1 \gt$
Work Step by Step
Our aim is to determine the maximum rate of change of $f(x,y)$.In order to find this, we have : $D_uf=|\nabla f(x,y)|$
Given: $f(s,t)=\sin (xy)$
$\nabla f(x,y)=\lt y \cos xy,x\cos xy \gt $
From the given data, we have $f(x,y)=f(1,0)$
Thus, $\nabla f(1,0)=\lt 0,\cos (0) \gt=\lt 0,1 \gt$
or, $|\nabla f(0,1)|=\sqrt{0^2+ 1^2}=1$
Therefore, the maximum rate of change of $f(x,y)$ and the direction is: $1, \lt 0,1 \gt$
