The directional derivative of a real-valued function 
at a point 
in the direction
of a nonzero vector 
is the limit
 |
(1)
|
when this limit exists, where 
is the corresponding unit
vector. If 
is differentiable at 
, then the chain rule gives
 |
(2)
|
where 
is the gradient and 
is called "nabla" or
"del" (Lebl 2026, pp. 40-42). Existence of
the partial derivatives alone does not justify
this gradient formula. All directional derivatives can
exist at a point where 
is not differentiable (Lebl
2026, pp. 44-45).
For a differentiable function, directional differentiation is also written in the notation
 |  |  |
(3)
|
 |  |  |
(4)
|
where 
denotes a unit vector in any given direction and 
denotes a partial derivative.
Let 
be a unit vector in Cartesian
coordinates, so
 |
(5)
|
then
 |
(6)
|
