Mathematics · Grades 11–12
Introductory calculus: interpret the result · Revision card
A derivative describes local rate; a definite integral describes signed accumulation.
Understand the idea
Rate and accumulation
A derivative describes local rate; a definite integral describes signed accumulation.
Use conditions
Differentiability implies continuity, but continuity alone does not imply differentiability.
Keep units
Derivative units are output per input; integral units multiply output and input units.
Worked examples
For s(t) = t² + 3t, find s′(2).
s′(t) = 2t + 3, so s′(2) = 7.
7
If s is distance in metres and t is time in seconds, the units of ds/dt are
The derivative compares change in distance with change in time.
metres per second
Explain without looking
- A velocity stays at 6 m/s throughout an interval. What is its derivative with respect to time on that interval?
Further reading and review references
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Preview material · independent educational review required. Grade bands are approximate, not official curriculum alignment.