# Tangent plane

to a surface $S$ at a point $M$

The plane passing through $M$ characterized by the property that the distance from this plane to a variable point $M_1$ of $S$ as $M_1$ approaches arbitrarily close to $M$ is infinitesimally small as compared to the distance $MM_1$. If $S$ is given by an equation $z=f(x,y)$, then the equation of the tangent plane at a point $(x_0,y_0,z_0)$, where $z_0=f(x_0,y_0)$, has the form

$$z-z_0=A(x-x_0)+B(y-y_0)$$

if and only if $f(x,y)$ has a total differential at the point $(x_0,y_0)$. In this case, $A$ and $B$ are the values of the partial derivatives $\partial f/\partial x$ and $\partial f/\partial y$ at $(x_0,y_0)$.