Given a function $ f:(a,b) \to E $ where $E$ is a Banach space, then clearly when $f$ is differentiable, the derivative $ f^\prime(x) $ can be seen as the tangent vector to the curve in $E$ graphed by $f$, at the point $f(x)$.
But the book I use says that value of the Frechet derivative of $f$ acting on $1$ i.e. $ f^\prime(x)(1) $ or $ Df(x)(1) $ is also the same as above definition.
I am confused. The infinitesimal "h" here is $1$. But isnt "h" supposed to be as small as possible and moreover it maybe that $ |b-a| < 1 $ ? How is this interpretation brought about?