Here we review the concept of the gradient and the directional derivative and prove the formula for the directional derivative discussed earlier.
Let f : U → ℝ where U is an open subset of ℝ^{n} and suppose f is differentiable on U. Thus if x ∈ U,
 (21.16) 
Now we can prove the formula for the directional derivative in terms of the gradient.
Proof:


Now lim_{t→0}

as claimed. ■
Example 21.9.2 Let f

Note this vector which is given is already a unit vector. Therefore, from the above, it is only necessary to find ∇f

Therefore, ∇f