There are many applications that can be interpreted as a sum of two functions. If, for example, you are standing in an elevator going up at speed v relative to the ground. Your height above the ground is f(t)= vt. You drop a ball which then has distance below your hand g(t)= -gt^2/2. The height of the ball above the ground is f(t)+ g(t)= -gt^2/2+ vt.

That can also be used the other way. If I have a problem that involves, for whatever reason, the function h(x)= 3x^4+ sin(x) and want to differentiate it, rather than having to memorize a "rule" for differentiating a power of xplusa trig function, I can think of it as the sum of f(x)= 3x^4 and g(x)= sin(x) and differentiate each separately.