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:

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

$\ds $

$\ds \frac 1 h \paren {\lambda \map f {\xi + h} + \mu \map g {\xi + h} - \lambda \map f \xi - \mu \map g \xi}$

$\ds $

$=$

$\ds \lambda \paren {\frac {\map f {\xi + h} - \map f \xi} h} + \mu \paren {\frac {\map g {\xi + h} - \map g \xi} h}$

$\ds $

$\to$

$\ds \lambda D \map f \xi + \mu D \map g \xi$

as $h \to 0$

The result follows from the definition of derivative.

$\blacksquare$