The sum and difference rule for differentiable equations states:
The sum (or difference) of two differentiable functions is differentiable and [its derivative] is the sum (or difference) of their derivatives.
$$\frac{\text{d}}{\text{d}x}[f(x) + g(x)] = f'(x) + g'(x)$$ $$\frac{\text{d}}{\text{d}x}[f(x) - g(x)] = f'(x) - g'(x)$$
What is the proof of this rule?