Source: /cirosantilli/total-derivative

= Total derivative
{wiki}

The total derivative of a function assigns for every point of the domain a linear map with same domain, which is the best linear approximation to the function value around this point, i.e. the tangent plane.

E.g. in 1D:
$$
Total derivative = D[f(x_0)](x) = f(x_0) + \pdv{f}{x}(x_0) \times x
$$
and in 2D:
$$
D[f(x_0, y_0)](x, y) = f(x_0, y_0) + \pdv{f}{x}(x_0, y_0) \times x + \pdv{f}{y}(x_0, y_0) \times y
$$