Derivatives as rates of change

Read and connect

1. Read: Defining Average and Instantaneous Rates of Change at a Point →

   Build a finite instantaneous rate from either form of the limiting secant quotient.

2. Read: Defining the Derivative of a Function and Using Derivative Notation →

   Connect the derivative function, derivative notation, and a tangent line at a specified point.

3. Read: Estimating Derivatives of a Function at a Point →

   Estimate a derivative from nearby data or a tangent, and retain approximation and unit labels.

4. Read: Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist →

   Check continuity and one-sided slopes to distinguish a smooth point, corner, cusp, or vertical tangent.

5. Read: Interpreting the Meaning of the Derivative in Context →

   Interpret a signed derivative with the quantity, input value, and output-per-input units.

6. Connect: Straight-Line Motion: Connecting Position, Velocity, and Acceleration →

   Later connection: apply the rate interpretation to calculated motion, direction, and changing speed.

   Before this connection: Differentiation tools; First and second derivatives.

7. Connect: Rates of Change in Applied Contexts Other Than Motion →

   Later connection: calculate and interpret applied rates after learning the required derivative rules.

   Before this connection: Differentiation tools; Chain rule and exponential derivatives.

8. Connect: Connecting Position, Velocity, and Acceleration of Functions Using Integrals →

   Later connection: recover motion from rates and distinguish displacement from distance.

   Before this connection: Accumulation and definite integrals; Antiderivatives and integration tools.