Sum Rule for Derivatives/Proof 2

Theorem

Let $\map f x, \map j x, \map k x$ 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 $\map f x = \map j x + \map k x$.

Then $f$ is differentiable at $\xi$ and:

$\map {f'} \xi = \map {j'} \xi + \map {k'} \xi$

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

$\forall x \in I: \map {f'} x = \map {j'} x + \map {k'} x$

Proof

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

$\blacksquare$