The Total Differential

The partial derivatives describe only the rate of change in a “single direction”, whereas the total differential describes the change brought about by a small increment as a whole — it writes the increment of the function as the sum of two (or more) increments plus a higher-order remainder.

The core of this chapter is the condition for differentiability: the function is differentiable at a point if and only if it can be approximated by a linear map plus a negligible remainder. This statement is considerably stricter than its one-variable counterpart, and it is the foundation of everything that follows. Geometrically, the total differential corresponds to the tangent plane of the surface — a small change in the value of the function equals the vertical drop of the tangent plane.

Differentiability implies continuity and implies the existence of the partial derivatives, but neither of the converses holds. Keeping this one-way implication clear in your head will save you a lot of wrong guesses.