Linear Combination of Derivatives

Theorem
Let $\map f x, \map g x$ be real functions defined on the open interval $I$.

Let $\xi \in I$ be a point in $I$ at which both $f$ and $g$ are differentiable.

Then:
 * $\map D {\lambda f + \mu g} = \lambda D f + \mu D g$

at the point $\xi$.

It follows from the definition of derivative that if $f$ and $g$ are both differentiable on the interval $I$, then:


 * $\forall x \in I: \map D {\lambda \map f x + \mu \map g x} = \lambda D \map f x + \mu D \map g x$

Proof
The result follows from the definition of derivative.