Lesson 4 of 18Article20 min
The Derivative: Instantaneous Rate
Average rate uses a chord (secant). Instantaneous rate uses the tangent — the limit of secant slopes as the second point slides toward the first.
From secant to tangent
Average rate uses a chord (secant). Instantaneous rate uses the tangent — the limit of secant slopes as the second point slides toward the first.
If the limit fails (different left/right slopes, vertical tangent, cusp), the derivative does not exist at that point — even if the function is continuous.
What f′ tells you
- f′(x) > 0 → f increasing near x.
- f′(x) < 0 → f decreasing near x.
- f′(x) = 0 → horizontal tangent — candidate for local max/min.
- Units: if f is metres and x is seconds, f′ is metres per second.
Worked example from definition
f(x) = x². [ (x+h)² − x² ] / h = (2xh + h²)/h = 2x + h → as h→0, f′(x) = 2x.