Differentiation is linear

Statement

The following are true:

Differentiation is additive, or derivative of sum is sum of derivatives: If $f$ and $g$ are functions that are both differentiable at $x = x_{0}$ , we have:

$\frac{d}{d x} [f (x) + g (x)]_{x = x_{0}} = f^{'} (x_{0}) + g^{'} (x_{0})$

or equivalently:

$(f + g)^{'} (x_{0}) = f^{'} (x_{0}) + g^{'} (x_{0})$

In point-free notation:

$(f + g)^{'} = f^{'} + g^{'}$

Constants (also called scalars) can be pulled out of differentiations: If $f$ is differentiable at $x = x_{0}$ and $λ$ is a real number, then:

$\frac{d}{d x} [λ f (x)] |_{x = x_{0}} = λ f^{'} (x_{0})$

Fill this in later