The second derivative is the derivative of the derivative — measures curvature, or in physics, acceleration.
Drag the point along the curve
y = x·xat x = 1: slope ≈ 2
What f″ tells you
- Sign of f″ — concavity (up if +, down if −).
- f″ changes sign — an inflection point.
- f′ = 0 and f″ ≠ 0 — that critical point is a max (f″ < 0) or min (f″ > 0).
- In physics — if f(t) is position, f″(t) is acceleration.
Your turn
If s(t) = 5t² − 3t metres, find the acceleration.
Notation: f″(x), d²y/dx², or y″. The little 2 isn't a power — it means 'differentiate twice'.
If position is f(t), velocity is f'(t), acceleration is f''(t).