Differentiation on Polynomials is Linear Operator

Theorem
Let $\map P \R$ be the vector space of all polynomial functions on the real number line $\R$.

Then the differentiation operator $D$ on $\map P \R$ is a linear operator.

Proof
Let $\map f x, \map g x$ be real functions which are differentiable on $\R$.

Then from Linear Combination of Derivatives:
 * $\forall x \in I: \map D {\lambda \map f x + \mu \map g x} = \lambda D \map f x + \mu D \map g x$

It follows from Real Polynomial Function is Differentiable that $\lambda D \map f x + \mu D \map g x$ is differentiable on $\R$.

The result follows from Condition for Linear Transformation.