total derivative
In one variable the derivative is a number — the slope of the tangent line. In several variables the honest analogue is not a number but a whole linear map: the best flat (linear) approximation of the function near a point. Where single-variable calculus hands you a slope, multivariable calculus hands you a linear transformation that tilts space.
Formally, f: R^n -> R^m is (totally) differentiable at a with total derivative Df(a) — a linear map from R^n to R^m — if f(a + h) = f(a) + Df(a) h + r(h) where the remainder satisfies |r(h)| / |h| -> 0 as h -> 0. The total derivative is unique when it exists, and its matrix in the standard bases is exactly the Jacobian matrix of partial derivatives.
This is the correct, coordinate-free notion of the derivative. It captures the geometric content — a tangent plane, a local linearization — that mere partial derivatives miss. From it the chain rule reads as composition of linear maps (matrix multiplication), and theorems like the inverse and implicit function theorems flow naturally.
For f(x, y) = x^2 + y^2 at the point (1, 1): Df(1,1)(h, k) = 2h + 2k, the linear map with matrix [2, 2]. Then f(1 + h, 1 + k) = 2 + 2h + 2k + (h^2 + k^2), and the remainder h^2 + k^2 vanishes faster than sqrt(h^2 + k^2).
The total derivative as the linear part of the increment.