Once you can differentiate a few simple pieces, you can differentiate any sum of them — just differentiate each piece and add the results. The derivative of a sum is the sum of the derivatives:

And because a difference is just a sum with a minus sign, the same holds for subtraction:

In words: differentiate term by term. You never have to deal with a whole expression at once — break it into pieces, differentiate each one, and reassemble.

Why it works: slopes just add

The derivative measures a slope — a rate of change. If at some instant f is climbing at 3 units per step and g is climbing at 2, then their sum f + g climbs at 3 + 2 = 5. Stacking one curve on top of the other simply adds their heights at every point, so it adds their slopes too.

Below, the faint curves are f(x) = \sin(x) and the gentle line g(x) = \tfrac{1}{2}x; the bold curve is their sum f(x) + g(x). At every x the bold height is the two faint heights added — and its slope is their two slopes added.

A worked example

Suppose h(x) = x^2 + 3x. This is a sum of two terms, so we handle each on its own. Using the power rule on x^2 and the constant multiple rule on 3x:

Now just add the two results back together:

A difference is no different. For x^2 - 3x we subtract instead: 2x - 3.

Here is that example drawn out — h(x) = x^2 + 3x (solid) beside its derivative h'(x) = 2x + 3 (dashed). Notice the dashed line crosses zero exactly where the solid curve bottoms out and stops falling.

It chains across many terms

The rule isn't limited to two terms — apply it repeatedly and a long sum comes apart into as many pieces as you like. Each term gets differentiated independently and the + and - signs carry straight through:

That single idea is the engine behind differentiating any polynomial — you'll put it together with the power rule next. First, see Sal work through the basic rules: