In differential geometry I'm told that directional derivatives can be interpreted as tangent vectors and I'm trying to build some intuition for this using simple cases. If I take a map $f: \Bbb R^2 \to \Bbb R$, $f(x,y)=2-x^2-y^2$ and a vector $v=\langle \frac{1}{\sqrt2}, -\frac{1}{\sqrt2}\rangle$, then the directional derivative is the gradient dotted with $v$ $$D_vf= \nabla f \cdot v = -\sqrt2x + \sqrt2y.$$ I fail to see how is this a vector? It's a real number in $\Bbb R$? So in a sense a vector in $\Bbb R$, but the only thing it's tangent to is the real line?
I'm trying to build up to the tangent space which uses derivations which seem to be in a sense some kind of directional derivative?