total derivative

The partial derivatives each answer a one-direction question; the total derivative answers all directions at once. It is the single best linear approximation to a function near a point — the multivariable replacement for the tangent line. Where a one-variable derivative is a number (the slope), the total derivative of f at a point is a linear map, a machine that takes any displacement vector and returns the function's predicted change.

For a scalar function f(x, y), the total derivative is the row of partials [partial f / partial x, partial f / partial y]; feeding it a small step (dx, dy) gives df = (partial f / partial x) dx + (partial f / partial y) dy, the best linear estimate of the actual change in f. For a vector-valued map this row becomes the full Jacobian matrix. The defining property is that the error of this linear estimate shrinks faster than the size of the step: f(a + h) - f(a) - Df(a) h is o(|h|) as h -> 0. That little-o condition is exactly what 'differentiable' means in several variables, and it is genuinely stronger than merely having partial derivatives.

This is the object you actually want. The gradient is the total derivative dressed as a vector; the chain rule is composition of total derivatives (matrices multiply); error propagation, Newton's method, and the linearization of dynamical systems all run on the total derivative. Thinking of the derivative as a linear map rather than a pile of partials is the conceptual leap that makes the rest of multivariable and vector calculus click into a single coherent picture.