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:
 * $\displaystyle D_x \left({\sum_{i=1}^n f_i \left({x}\right)}\right) = \sum_{i=1}^n D_x \left({f_i \left({x}\right)}\right)$

Proof 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.