In one variable, differentiability means that a function looks increasingly like its tangent line when viewed close to a point. In several variables, the same idea becomes approximation by a linear map.

A scalar-valued function $f:\mathbb R^n\to\mathbb R$ is differentiable at a point $\mathbf a$ when there is a linear map $Df(\mathbf a)$ such that, for a small displacement $\mathbf h$,

$$f(\mathbf a+\mathbf h)

f(\mathbf a)+Df(\mathbf a)\mathbf h+o(|\mathbf h|).$$

The error becomes negligible compared with the size of $\mathbf h$ as $\mathbf h\to\mathbf0$.

The tangent-plane idea

For $z=f(x,y)$, differentiability means that the surface near $(a,b)$ is well approximated by a plane. Using the partial derivatives, that linear approximation is

$$f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b).$$

The corresponding graph is the tangent plane.

Partial derivatives are not enough

A function may have partial derivatives in every coordinate direction at a point and still fail to be differentiable there. Partial derivatives inspect a few directions; differentiability requires one linear approximation that works for all sufficiently small displacements.

A useful sufficient condition

If the first partial derivatives exist in a neighborhood of the point and are continuous there, then the function is differentiable at that point.

Differentiability therefore ties the separate partial derivatives together into one local linear model of the function.