Considering the derivative focus point for $x=a$ allows one to consider the question of what value a linear function has for a change in $x$, $\Delta x = dx$.

To estimate $f(a + \Delta x)$ the value $f(a)$ is changed by $dy=f'(a) \Delta x = f'(a) dx$, so $f(a + \Delta x) \approx f(a) + dy = f(a) + f'(a) dx$.