Derivative Rules: Constant, Sum, Difference, and Constant Multiple

A derivative measures slope, and slope only cares about how steeply a graph rises or falls, not where it sits. Add 100 to a function and every point on its graph just shifts straight up by 100 — the graph looks identical otherwise, so the slope at every x is untouched. That is the constant rule: the derivative of any constant by itself is 0.

Multiplying a function by a constant is different: it stretches the graph vertically, and stretching a graph vertically stretches every slope by that same factor. If f(x) has slope m at some point, 3f(x) has slope 3m there, because every rise gets tripled while every run stays the same. That is the constant multiple rule. The sum and difference rules say you can differentiate a polynomial piece by piece: the slope of a sum is the sum of the slopes, so d/dx[u(x) + v(x)] = u'(x) + v'(x), and the same goes for a difference.

These four rules are what turn the power rule from a fact about single terms like xⁿ into a tool for entire polynomials. Once you can peel off constants, split sums apart, and drop constants that stand alone, differentiating any polynomial becomes a line-by-line routine rather than a limit computation. This is the bridge from the formal definition of the derivative, which you used to find derivatives from scratch, to the fast symbolic rules — product, quotient, chain — that come next.