Statement
The derivative of two functions is the derivative of the first function plus the derivative of the second function.
$$ \tfrac{d}{dx}\bigg(g(x) + j(x)\bigg) = g'(x) + j'(x) $$
Proof
Define function $ f $.
$$ f(x) = g(x) + j(x) $$
Use the definition of the derivative.
$$ f'(x) = \lim_{h \to 0}\left(\frac{f(x + h) - f(x)}{h}\right) $$
$$ f'(x) = \lim_{h \to 0}\left(\frac{g(x + h) + j(x + h) - (g(x) + j(x))}{h}\right) $$
$$ f'(x) = \lim_{h \to 0}\left(\frac{g(x + h) + j(x + h) - g(x) - j(x)}{h}\right) $$
Rearrange the terms and split the fraction.
$$ f'(x) = \lim_{h \to 0}\left(\frac{g(x + h) - g(x) + j(x + h) - j(x)}{h}\right) $$
$$ f'(x) = \lim_{h \to 0}\left(\frac{g(x + h) - g(x)}{h} + \frac{j(x + h) - j(x)}{h}\right) $$
Split the limit.
$$ f'(x) = \lim_{h \to 0}\left(\frac{g(x + h) - g(x)}{h}\right) + \lim_{h \to 0}\left(\frac{j(x + h) - j(x)}{h}\right) $$
Now two definitions of the derivative arised.
$$ f'(x) = g'(x) + j'(x) $$