Let $ f(x) = x+a$. Accordingly, $ f(x+h) =x+h+a$

By first principle,

$ f'(x) \lim\limits_{ h \to 0} \large\frac{f(x+h)-f(x)}{h}$

$ = \lim\limits_{ h \to 0} \large\frac{x+h+a-x-a}{h}$

$ = \lim\limits_{ h \to 0} \bigg( \large\frac{h}{h}\bigg)$

$ = \lim\limits_{ h \to 0} (1)$

$ = 1$