where
and so that
and
We can find by using the formula for the increment .
where depends on and as .
Hence,
The term does not depend on
. As
and , and therefore
.
Hence,
Thus the directional derivative of f at (x,y) in the direction of the unit
vector is the dot product of the gradient of f at that point
and the unit vector.
In short, is the component of
in the direction of .
If f is differentiable, and we find the gradient at a given point, we can calculate the directional derivative of f at that point in any direction .
If , then and similarly so that in the case the directional derivative give rate of f in the i and j directions and coincide with the partial derivatives.
The concept of the directional derivative and its relation to the gradient can be extended to the
functions three (or more) variables. If w=f(x,y,z) is a
differentiable function, then the directional derivative at (x,y,z) in the
direction of the unit vector is
where are the direction cosines a line through
x,y,z in the direction .
In vector notation
and
so that
and
and
Thus
where
depends on
and
and tends to zero as
tend to (0,0,0).
Since f is differentiable, the same mode of proof as for a function of two variables, shows
that
Again the directional derivative is the dot product of the gradient vector and the unit vector
.