Grade 12 · Mathematics
Definition
Why it matters
Instantaneous rate of change.
Advanced20m readingPrerequisite: None
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.
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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.