Derivative
Definition
The instantaneous rate of change of something — how fast it is changing at one exact moment. The derivative of position is velocity; the derivative of velocity is acceleration.
Symbol: ·
Technical definition
The limit of the average rate of change as the interval shrinks to zero: f'(x) = lim(h→0) [f(x+h) − f(x)]/h. Geometrically, the slope of the tangent to the curve at a point; physically, the instantaneous rate of change.
Example
If position is x = 4t^2, the derivative dx/dt = 8t gives velocity at any instant — at t = 3 s the object moves at 24 m/s. Differentiating once more gives the constant acceleration, 8 m/s^2.
The derivative is not the same as the average rate of change (Δy/Δx over an interval) — it is the limit as the interval vanishes. For curved graphs the average slope between two points differs from the instantaneous slope at either point, and physics usually needs the instantaneous one.