Span of Exponential of a x by Cosine of b x and Exponential of a x by Sine of b x is preserved under Differentiation wrt x

Theorem
Let $a, b, x, \alpha_1, \alpha_2 \in \R$ be real numbers.

Denote $\ds f_1 = \map \exp {a x} \map \cos {b x}$, $f_2 = \map \exp {a x} \map \sin {b x}$.

Let $\map \CC \R$ be the space of continuous real-valued functions.

Let $\tuple {\map {\CC^1} \R, +, \, \cdot \,}_\R$ be the vector space of continuously differentiable real-valued functions.

Let $S = \set {f_1, f_2} \subset \map {\CC^1} \R$ be a vector space.

Let $\span S$ be the span of $S$.

Then differentiation $x$ preserves $\span S$.