Answer
$\nabla f =-i+j$ starting at initial point (2,1) on the level curve $-1 =\space y-x$
Work Step by Step
Our aim is to take the first partial derivative of the given function $f(x,y)$ with respect to $x$, by treating $y$ and $x$ as a constant, and vice versa:
$f_x=\dfrac{\partial }{\partial x}(y-x)$
and $f_x(2,1)=-x^{0}=-1|_{(2,1)}$
and $f_y=\dfrac{\partial }{\partial x}(y-x)$
and $f_y(2,1)=y^{0}=1|_{(2,1)}$
We need to write the gradient vector that extends 1 unit to the left and 1 unit up.
$\nabla f = \lt -1,1 \gt =-i+j$ and $f(2,1)=1-2=-1$
The equation of the level curve is equal to $-1 =\space y-x$
Thus, we have $\nabla f =-i+j$ starting at initial point (2,1) on the level curve $-1 =\space y-x$
