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:

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

General Result
Let $$f_1 \left({x}\right), f_2 \left({x}\right), \ldots, f_n \left({x}\right)$$ be real functions all differentiable as above.

Then $$D_x \left({\sum_{i=1}^n f_i \left({x}\right)}\right) = \sum_{i=1}^n D_x \left({f_i \left({x}\right)}\right)$$.

Proof 1
$$ $$ $$ $$ $$ $$ $$

Proof 2
It can be observed that this is an example of a Linear Combination of Derivatives with $$\lambda = \mu = 1$$.

Proof of General Result
Follows directly by induction.