Linear Combination of Derivatives

Theorem
Let $f \left({x}\right), g \left({x}\right)$ 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:
 * $D \left({\lambda f + \mu g}\right) = \lambda D f + \mu D g$

at the point $\xi$.

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


 * $\forall x \in I: D \left({\lambda f \left({x}\right) + \mu g \left({x}\right)}\right) = \lambda D f \left({x}\right) + \mu D g \left({x}\right)$