SEC2_GYM · Calculus · CALC.DERIV.CONCEPT

Derivative as local rate of change

The derivative measures instantaneous rate of change — geometrically, the slope of the tangent.

Practice this skill →

Intuition

Understand first, compress later.

A secant through two nearby points becomes the tangent as the distance between the points tends to zero.

Switch representations

Secant becomes tangentf(x) = x², x = 1

curve fsecant Δx = 1.2secant Δx = 0.6secant Δx = 0.2tangent, slope 2point of tangency

All three secants pass through (1, 1). As Δx shrinks (1.2 → 0.6 → 0.2) the secant slope moves 3.2 → 2.6 → 2.2 — towards the tangent slope f′(1) = 2. That limit is the derivative.

Formal view

, if this limit exists.

Why does it work?

The difference quotient is the average rate of change over an interval. Taking the limit turns it into a local rate at a point.