Defining the Derivative of a Function and Using Derivative Notation

A single slope only describes one point. But a curve bends differently everywhere along its length, so you need a rule that hands back the slope at any x you ask for. That rule is the derivative, f'(x), a brand new function built from f(x). It exists because 'the slope of f' is not one number — it's a whole pattern worth capturing.

The tool for building that pattern is the difference quotient: [f(x+h) - f(x)] / h. This measures the slope of a secant line through two points on the curve, x and x+h, close together. As h shrinks toward 0, the secant line rotates and settles onto the tangent line at x. So f'(x) = lim(h→0) [f(x+h) - f(x)] / h. Because x is still a variable in this expression, the limit sweeps out an entire new curve, one slope value for every x, not just one number.

This definition is the foundation everything else in differentiation rests on: the power rule, product rule, and chain rule are all shortcuts that give the same answer as this limit, just faster. Notation matters here too. Leibniz notation, dy/dx, writes the derivative as if it were a fraction of two tiny changes, because it's built from the fraction [f(x+h)-f(x)]/h — but it is one limiting value, not two separate quantities dy and dx sitting around on their own. That distinction becomes important later, when related rates and differential equations seem to split dy/dx apart; the manipulation works, but not because dy and dx are ordinary numbers.