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:

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

Proof
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