1. Definition of a Derivative
Keeper 13
Honors Calculus

2. Understanding Functions
Label the following on the picture below: 𝑓 𝑥 , 𝑓 𝑥+ℎ , ℎ, 𝑓 𝑥+ℎ −𝑓(𝑥) and a segment whose slope represents 𝑓 𝑥+ℎ −𝑓 𝑥 ℎ and lim ℎ→0 𝑓 𝑥+ℎ −𝑓 𝑥 ℎ

3. Slope of the Secant Line
𝑚 𝑠𝑒𝑐 = 𝑓 𝑥+ℎ −𝑓 𝑥 ℎ This formula represents the AVERAGE rate of change

4. Slope of the Tangent Line
𝑚 𝑡𝑎𝑛 = lim ℎ→0 𝑓 𝑥+ℎ −𝑓 𝑥 ℎ This formula represents the INSTANTANEOUS rate of change

5. Compare and Contrast the Average Rate of Change with the instantaneous rate of Change (derivative).
Similarities
Differences

6. Derivative Slope of a tangent line Rate of change at 1 point
Instantaneous rate of change
Denoted by 𝑦 ′ , f ′ x , or 𝑑𝑦 𝑑𝑥

7. Definition of a Derivative
𝑑𝑦 𝑑𝑥 = lim ℎ→0 𝑓 𝑥+ℎ −𝑓 𝑥 ℎ

8. Find the Derivative using the definition
1. 𝑓 𝑥 =−3𝑥+2

9. Find the Derivative
2. 𝑓 𝑥 = 3𝑥 2 −2𝑥+4

10. Find the Derivative
3. 𝑓 𝑥 = 1 𝑥+1

11. Find the Derivative
4. 𝑓 𝑥 = 𝑥−2