Sum Rule for Derivatives

Theorem
Let $$f \left({x}\right), j \left({x}\right), k \left({x}\right)$$ be real functions defined on the open interval $$I$$.

Let $$\xi \in I$$ be a point in $$I$$ at which both $$j$$ and $$k$$ are differentiable.

Let $$f \left({x}\right) = j \left({x}\right) + k \left({x}\right)$$.

Then $$f^{\prime} \left({\xi}\right) = j^{\prime} \left({\xi}\right) + k^{\prime} \left({\xi}\right)$$.

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

$$\forall x \in I: f^{\prime} \left({x}\right) = j^{\prime} \left({x}\right) + k^{\prime} \left({x}\right)$$.

Proof
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