find the eqn of the tangent plane to the surface

z=f(x,y)=x^2 +4y^2

at the point (a,b,a^2 +4y^2)

What I don't understand is how the normal to the plane at point a,b is geven by:

n=(fx,fy,-1)

would fx not be the "rate of change with respect to x" and not the NORMAL to the plane at that point?