- Given the plane curve $y=f(x)=\cos(x)$, the function $f'(x)=-\sin(x)$ models the slope of the tangent line to $f$ at each point.
- Given the surface $z=f(x,y)=x^2+y^2$, the functions $z_x=2x$ and $z_y=2y$ model the slopes of the tangent lines to the $x$ and $y$ curve-slices of $f$ at each point.
But neither $z_x$ nor $z_y$ are the analogous "derivative of $f$ as a function." That is, neither models the slope of the tangent plane to the surface $f$ at every point--like the effect of the single variable derivative.
My question: is there a function which models the slope of the tangent plane to a surface at every point?
My guess: The differential does give us the whole picture as far as rate of change in each coordinate direction: $dz=2xdx+2ydy$. But $dz$ is a differential 1-form i.e. a covector field. If my guess is correct, that $dz$ is the analogous "derivative of $f$ as a function," does it have a graph? Can $dz$ be realized as a surface?
In the loosest of phrasing, similar to how the derivative of a curve is a curve, is the derivative of a surface a surface?