I have question about presentation of directional derivative.

Sometimes I see directional derivates and gradient to be represented as a function but I think that it is convenient to use "vector presentation" instead.

My question is that is there some cases and/or good reasons why should use functional presentation instead of vector?

To make my point clear I wrote an example what I mean about "presentations".

Simply I mean the presentation of vector, using comma versus using i and j components

There is one example twice, for "function presentation" and for "vector presentation".

Let we have a function: and we calculate gradient and directional derivative of function .

Function presentation:

Gradient (I include all phases here):

Next we calculate directional derivative of function f at this point (2,-1) to the direction of

The vector has magnitude . The unit vector in the direction is thus

Directional derivative is then

and plugging the point (2,-1) we get

Vector presentation

Next we calculate directional derivative of function f at this point (2,-1) to the direction of

Let's plugin the point to the Gradient:

The vector has magnitude . The unit vector in the direction is thus

Directional derivative is then

Calculations above are the same no matter when we plugin the point (2,-1).

I think that representing the gradient as a vector is more clear than function.

Now, I will repeat my question:

Is there some cases and/or good reasons why should use functional presentation (i,j components) instead of vector presentation in points (x,y)?