Calculus 12 · Mathematics · Lesson 3.1
Definition
- Level
- Grade 12
- Reading time
- 14 min reading
- Review
- Last reviewed 2026-09-03
- Standard
- Accuracy standard
- Errata
Abstract
Instantaneous rate of change.
What is a derivative?
Short answer
A derivative measures the instantaneous rate of change of a function — the slope of the tangent line to its graph at a point. It is defined as the limit of the average rate of change over an interval as the interval shrinks to zero: f′(x) = lim(h→0) [f(x+h) − f(x)] / h. Geometrically it is a slope; physically it is a rate such as velocity.
Concept preview
What you will be able to do
- 01Interpret the derivative as an instantaneous rate of change and as the slope of a tangent line.
- 02Calculate the derivative at a point from the limit definition of the difference quotient.
- 03Derive the derivative function from first principles for polynomial, simple rational and root functions.
- 04Determine where a function fails to be differentiable, distinguishing corners, cusps, vertical tangents and discontinuities.
- 05Justify why differentiability implies continuity and diagnose a flawed first-principles argument.
Every applied question about change eventually asks not for an average but for an instant. A car's trip meter reports the average speed over a whole journey, yet the speedometer reads the speed at this very moment; a population report gives the average growth over a year, yet a biologist may want the rate of growth on a single day. The derivative is the mathematical tool that turns an average rate of change, computed over an interval, into an instantaneous rate of change, defined at a single point. The whole idea rests on a simple manoeuvre you already know how to make precise: compute the average rate over a shrinking interval and take the limit as the interval closes to nothing. The derivative is that limit, and the rest of this lesson is really an unpacking of that one sentence.
School Plan
Definition
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Related topics
Frequently asked questions
- What does the derivative tell you geometrically?
- The slope of the line tangent to the curve at that point — how steeply the function is rising or falling there.
- What is the difference between average and instantaneous rate of change?
- Average rate is the slope over an interval; instantaneous rate is the limit of that slope as the interval shrinks to zero.