Foundations of Differential Calculus
1Introduction
This lecture presents differentiation in three stages: defining rates of change through difference quotients, interpreting differentiation as local linear approximation, and computing derivatives through differentiation rules.
2Learning Sequence
- At each point where the difference-quotient limit exists, define the derivative at that point, and form the derivative function whose domain is the set of those points.
- At an interior point, characterize differentiability from two equivalent perspectives: equality of the left and right limits of the difference quotient, and a local linear approximation whose remainder satisfies , meaning .
- Use local linear approximation to replace a curve near a reference point by a line.
- Select differentiation rules according to the additive, multiplicative, quotient, or composite structure of the function.
3Quantities That Change and Interpretations That Are Retained
| Perspective | Change | Interpretation retained |
|---|---|---|
| Difference quotient | When the limit exists, an average rate of change between two points tends to an instantaneous rate at one point | At a differentiability point, the limit of the difference quotient gives the slope of the tangent line |
| Local linear approximation | A curve is approximated by a line | The value and slope at the reference point |
| Differentiation rules | A function expression is processed by decomposing its additive, product, quotient, and composite structure | The interpretation grounded in the definition of the derivative |