Total derivative

The best linear approximation to $f$ at a point -- a single matrix, $Df(\mathbf{a})$ , not a list of separate partials.

The total derivative $Df(\mathbf{a})$ of $f:\mathbb{R}^n\to\mathbb{R}^m$ at $\mathbf{a}$ is the linear map for which $f(\mathbf{a}+\mathbf{h}) \approx f(\mathbf{a}) + Df(\mathbf{a})\mathbf{h}$ with error shrinking faster than $|\mathbf{h}|$ . Existing partial derivatives alone don't guarantee this map exists; a genuine total derivative is a stronger, coordinate-free notion of differentiability.

Examples

For $f(x,y)=(x^2y,\ x+y)$ , $Df(x,y)$ is the $2\times2$ Jacobian matrix of first partials, evaluated at the point.