Unit 2
Definition of Derivatives
5–7% AB & BCCED Unit 2
Definition of the Derivative
Average vs. instantaneous rate of change, the limit definition, notation, and differentiability.
Average vs. Instantaneous Rate of Change
Average Rate of Change: the slope of the secant line between two points:
Average Rate of Change
Instantaneous Rate of Change: the slope of the tangent line at a single point:
Instantaneous Rate of Change
This limit IS the derivative at .
The Limit Definition of the Derivative
Form 1 (h → 0)
Form 2 (x → a)
Both forms compute the same thing: the slope of the tangent line. Form 1 gives the derivative as a function; Form 2 gives the derivative at a specific point.
Derivative Notation
For , the derivative can be written as:
All mean the same thing. Leibniz notation () is especially useful for chain rule and related rates.
Differentiability and Continuity
Theorem
If is differentiable at , then is continuous at .
⚠ Watch Out
The converse is FALSE. Continuity does NOT guarantee differentiability.
A function is NOT differentiable at points where there is a:
CornerCuspVertical TangentDiscontinuity
→ Tip
Classic example: is continuous at but NOT differentiable there (corner).
AP® Calculus AB & BC · Unit 2 Overview · Mr. Brantley