what is a curve ? Is the concept of derivative limited to curves only? I am trying to understand derivative and I want to know intuitive and rigorous definitions for a curve and if derivative is lmited only to curves or not..

A curve is the graph of an application \begin{align*}\gamma :A\subset \mathbb R&\longrightarrow \mathbb R^n\\ t&\mapsto (\alpha_1(t),...,\alpha_n(t))\end{align*}
where \begin{align*}
\alpha_i:A&\longrightarrow \mathbb R\\
t&\mapsto \alpha_i(t).
\end{align*}
And yes, derivative are limited to curve. For a surface or any other space... you will talk about differential and not derivative. For a curve, the derivative and the differential coincident. 
