Geometry of Partial Derivatives

This shows the graph of a function $z=f(x,y)$. We have marked a point $(x,y,z)=(a,b,0)$ in the $xy$-plane, and a bar going upwards from there to the point $(a,b,f(a,b))$ on the surface where $z=f(x,y)$.

Here we have cut across the surface with a plane of the form $y=b$, so we are holding $y$ constant and allowing $x$ to vary. The intersection of the plane with the surface is a curve, and we have also drawn a tangent line to that curve. The slope of that tangent line is $f_x(a,b)$ or (in alternative notation) the value of $\partial f/\partial x$ at $(a,b)$.

Here we have cut across the surface with a plane of the form $x=a$, so we are holding $x$ constant and allowing $y$ to vary. The intersection of the plane with the surface is a curve, and we have also drawn a tangent line to that curve. The slope of that tangent line is $f_y(a,b)$ or (in alternative notation) the value of $\partial f/\partial y$ at $(a,b)$.

Here we have drawn a small piece of the tangent plane to the surface $z=f(x,y)$ at the point $(a,b,f(x,b))$. This plane contains the two tangent lines with slopes $f_x(a,b)$ and $f_y(a,b)$.