(Note, that since the d's are operators rather than variables, they do not cancel.) Here, and are infintessimaly small changes in and respectively, called differentials.

The gradient of a function is also known as its derivative. Commonly
used notation includes for the derivative with respect to
and for the derivative with respect to .
(Note, that derivate and differential are *not* the same thing.
The *derivative* of with respect to is
, but the *differential* of is .)

Using the definition of the derivative above, we can deduce the gradient function of any function for which it is defined. For example, for the function ,

So to find the gradient of at the point , we substitute the coordinate into the gradient function to obtain . The geometric significance of this is that at this point, for every unit change in , we need to increase by six units to remain on the tangent to the curve at this point.

Differentiation is a linear transformation. That is to say

where