Question

Find the slope of the surface in the x and y-directions at the given point. h(x,y) = x2 – y2 (-3, 2, 5)

Answer 1

If $h(x,y)=x^2-y^2$, then you can find partial derivatives: 
$\frac{dh}{dx} =2x \\ \frac{dh}{dy} =-2y$.

At point A(-3,2,5) (this is the point which lies on the surface, because $5=(-3)^2-2^2=9-4=5$) these derivatives have values:

$\frac{dh}{dx} |_A=2\cdot (-3)=-6 \\ \frac{dh}{dy} =-2\cdot 2=-4$ -- the slopes at x and y- directions, respectively.