Subspace of Real Differentiable Functions

Theorem
Let $$\mathbb J = \left\{{x \in \R: a < x < b}\right\}$$ be an open interval of the real number line $$\R$$.

Let $$\mathcal D \left({\mathbb J}\right)$$ be the set of all differentiable real functions on $$\mathbb J$$.

Then $$\left({\mathcal D \left({\mathbb J}\right), +, \times}\right)_\R$$ is a subspace of the $\R$-vector space $$\left({\R^{\mathbb J}, +, \times}\right)_\R$$.