Linear Combination of Derivatives

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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.


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

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.