Some clarification needed on the Relation between Total Derivative and Directional Derivative I will consider here functions of several variables only.
If both directional derivative $D_{v}f(x)$  at  $x$  along  $v$   and  total derivative $D f(x)$ at  $x$  exist  then
$$D_{v}f(x)=Df(x)(v).$$
Existence of total  derivative ensures  that  of  directional derivative  in every direction but not the other way round. 
There are functions who have, at some point of the domain, directional derivative in every direction but not differentiable at that point i.e. the total derivative at that point does exist.
Now, all of my knowledge is theoretical. I cannot see the picture clearly, i.e.  the picture of the two kinds of derivative existing together, or one existing and not the other - how do these work?
I mean some geometrical interpretation for say $2$ or $3$ dimensional space would help.
I am so confused with this thing, I am not even sure if I have managed to convey my problem properly.
Please help with some clarification.
Thanks.

How about this?  The function $$f(x,y) = \begin{cases} 1 \text{ if $y = x^2$  and $x \neq 0$} \\ 0 \text{ otherwise}\end{cases}$$
Draw a picture.  You will see that every directional derivative exists and is equal to $0$, but the $x-y$ plane is not a good approximation for the function $f$ near $(0,0)$.
