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:

T o t a l d e r i v a t i v e = D [f (x_{0})] (x) = f (x_{0}) + \frac{\partial f}{\partial x} (x_{0}) \times x

(1)

and in 2D:

D [f (x_{0}, y_{0})] (x, y) = f (x_{0}, y_{0}) + \frac{\partial f}{\partial x} (x_{0}, y_{0}) \times x + \frac{\partial f}{\partial y} (x_{0}, y_{0}) \times y

(2)

Total derivative

ID: total-derivative