Linear Combination of Derivatives

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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:

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

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $		$\displaystyle \frac 1 h \left({\lambda f \left({\xi + h}\right) + \mu g \left({\xi + h}\right) - \lambda f \left({\xi}\right) - \mu g \left({\xi}\right)}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \lambda \left({\frac {f \left({\xi + h}\right) - f \left({\xi}\right)} h}\right) + \mu \left({\frac {g \left({\xi + h}\right) - g \left({\xi}\right)} h}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\to$	$\displaystyle \lambda D f \left({\xi}\right) + \mu D g \left({\xi}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\)	\(\displaystyle \frac 1 h \left({\lambda f \left({\xi + h}\right) + \mu g \left({\xi + h}\right) - \lambda f \left({\xi}\right) - \mu g \left({\xi}\right)}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \lambda \left({\frac {f \left({\xi + h}\right) - f \left({\xi}\right)} h}\right) + \mu \left({\frac {g \left({\xi + h}\right) - g \left({\xi}\right)} h}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\to\)	\(\displaystyle \lambda D f \left({\xi}\right) + \mu D g \left({\xi}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)