Why a function to be smooth requires more than one derivative? I don't understand the definition of smooth function like a function that is derivable infinite times.
If "smooth" means that the graph has not sharp corners it is sufficient only the existence of the first derivative.
 A: In mathematics, definitions are made objective so that they cannot be understood in different ways by different people. 
"Continuity" is quite opinion-based in its own right. Fortunately, results involving continuity do not rely on personal interpretation, because we make a covenant in advance to agree on what we mean by "continuity".
For the present question, the same idea applies.
A faithful suggestion is this: Familiarize yourself with the axiomatic thinking if you want to study mathematics seriously. Mathematics is not a real poem where the meaning of words are deliberately made open allowing different interpretation varying from person to person. 
A: You're mixing up the layman's definition with the math definition. Smooth means infinitely differentiable, having corners or not has nothing to do with it. 
Smooth means infinitely differentiable because "we say so".
Edit, as mrf says smooth can mean different things depending on contexts. But the point is that in each context it has a precise definition that only loosely has to do with an intuitive idea of "smooth".
A: The word "smooth" is very over-used. It tends to mean "as differentiable as I need right now". In some contexts, it's used for $C^\infty$ (derivatives of all order exist and are continuous), in some for $C^1$ (first order derivative exists and is continuous). In some situation it could mean even more complicated things, like belonging to some Sobolev space.
Check the book you are using.
