1. Alternate Definition of a Derivative
Section 2.1A
Calculus AP/Dual, Revised ©2018
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§2.1A: Alternate Definition of a Derivative

2. §2.1A: Alternate Definition of a Derivative
Alternate form of the Derivative Equation: 𝒇 ′ 𝒄 = 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 −𝒇 𝒄 𝒙−𝒄
This equation will allow the showing of the slope of the tangent line at a particular point
When investigating the relationship between differentiability and continuity
The existence of this limit requires that the one-sided limits exist and are equal: 𝐥𝐢𝐦 𝒙→ 𝒄 − 𝒇 𝒙 −𝒇 𝒄 𝒙−𝒄 = 𝐥𝐢𝐦 𝒙→ 𝒄 + 𝒇 𝒙 −𝒇 𝒄 𝒙−𝒄
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§2.1A: Alternate Definition of a Derivative

3. §2.1A: Alternate Definition of a Derivative
Review
Find the slope of the tangent line to the graph of the function 𝒇 𝒙 =𝟓− 𝒙 𝟐 , at the given point 𝟐, 𝟏
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§2.1A: Alternate Definition of a Derivative

4. §2.1A: Alternate Definition of a Derivative
Review
Find the slope of the tangent line to the graph of the function 𝒇 𝒙 =𝟓− 𝒙 𝟐 , at the given point 𝟐, 𝟏
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§2.1A: Alternate Definition of a Derivative

5. §2.1A: Alternate Definition of a Derivative
Example 1
Use the alternative form to find slope of the tangent line to the graph of the function 𝒇 𝒙 =𝟓− 𝒙 𝟐 , at the given point 𝟐, 𝟏
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§2.1A: Alternate Definition of a Derivative

6. §2.1A: Alternate Definition of a Derivative
Example 1
Use the alternative form to find slope of the tangent line to the graph of the function 𝒇 𝒙 =𝟓− 𝒙 𝟐 , at the given point 𝟐, 𝟏
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§2.1A: Alternate Definition of a Derivative

7. §2.1A: Alternate Definition of a Derivative
Example 2
Use the alternative form to find the derivative of 𝒇 𝒙 = 𝟏 𝒙 when 𝒄=𝟒
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§2.1A: Alternate Definition of a Derivative

8. §2.1A: Alternate Definition of a Derivative
Example 2
Use the alternative form to find the derivative of 𝒇 𝒙 = 𝟏 𝒙 when 𝒄=𝟒
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§2.1A: Alternate Definition of a Derivative

9. §2.1A: Alternate Definition of a Derivative
Example 3
Use the alternative form to find the derivative of 𝒇 𝒙 = 𝒆 𝒙 +𝟐𝒙 when 𝒄=𝟎
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§2.1A: Alternate Definition of a Derivative

10. §2.1A: Alternate Definition of a Derivative
Your Turn
Use the alternative form to find the derivative of 𝒇 𝒙 = 𝒙 𝟐 +𝟑, when 𝒄=−𝟐
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§2.1A: Alternate Definition of a Derivative

11. When I Say Derivative, You Say…
SLOPE
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§2.1A: Alternate Definition of a Derivative

12. §2.1A: Alternate Definition of a Derivative
Graphing a Derivative
When looking at a graph, identify what type of slope that each point has
If 𝒇 has a positive slope, 𝒇′ is increasing
If 𝒇 has a negative slope, 𝒇′ is decreasing
After establishing all of the slopes, determine the derivative graph by its slope.
In other words, plot points of the derivative onto the graph
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§2.1A: Alternate Definition of a Derivative

13. §2.1A: Alternate Definition of a Derivative
Steps
Establish any slopes of zero
Establish any steep and flat the slopes are
Plot the points
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§2.1A: Alternate Definition of a Derivative

14. §2.1A: Alternate Definition of a Derivative
Example 4
Sketch the derivative of the function given.
m = 1.5
Neg.
Pos. 5
1.5 3
0.5
Negative Slope
Positive Slope
(1.5, 5)
Dec.
Inc.
Decreasing
Increasing 5
1.5 3
0.5
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§2.1A: Alternate Definition of a Derivative

15. §2.1A: Alternate Definition of a Derivative
Example 5
Sketch the derivative of the function given.
Pos. Slope
Pos.
Pos. –3 –1
1/2 2
Neg.
Neg.
Inc.
Inc.
Inc. –3 –1
1/2 2
Dec.
Dec.
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§2.1A: Alternate Definition of a Derivative

16. §2.1A: Alternate Definition of a Derivative
Example 6
Sketch the derivative of the function given.
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§2.1A: Alternate Definition of a Derivative

17. §2.1A: Alternate Definition of a Derivative
Example 6
Sketch the derivative of the function given.
Increases
Increases + +
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§2.1A: Alternate Definition of a Derivative

18. §2.1A: Alternate Definition of a Derivative
Your Turn
Sketch the derivative of the function given.
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§2.1A: Alternate Definition of a Derivative

19. Direction of a Derivative from a Graph
A graph which is differentiable is continuous
A graph which is continuous, is not always differentiable
A graph is neither continuous or differentiable is discontinuous
Hole
Vertical Asymptotes
Jump discontinuities
Kinks/Cusps
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§2.1A: Alternate Definition of a Derivative

20. §2.1A: Alternate Definition of a Derivative
Special Cases
Differentiable functions are Continuous but continuous functions are not always Differentiable
Functions can not be differentiable when:
Sharp turns or “cusps” (i.e. 𝒇(𝒙)=|𝒙−𝟐|)
Vertical Tangents (i.e. 𝒇 𝒙 = 𝒙 𝟏/𝟑 )
Discontinuity
A function 𝒇 is said to be differentiable at 𝒙= 𝒄 if the derivative of 𝒇 exists at 𝒙=𝒄. For the derivative to exist at 𝒙=𝒄, the graph of 𝒇 must have local linearity.
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§2.1A: Alternate Definition of a Derivative

21. §2.1A: Alternate Definition of a Derivative
Example 7
Use the graph to determine the derivative of 𝒇 𝒙 = 𝒙−𝟐 . If it does not exist, explain why.
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§2.1A: Alternate Definition of a Derivative

22. §2.1A: Alternate Definition of a Derivative
Example 7
Use the graph to determine the derivative of 𝒇 𝒙 = 𝒙−𝟐 . If it does not exist, explain why.
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§2.1A: Alternate Definition of a Derivative

23. §2.1A: Alternate Definition of a Derivative
Example 7
Use the graph to determine the derivative of 𝒇 𝒙 = 𝒙−𝟐 . If it does not exist, explain why.
𝒇 is not differentiable at 𝒙=𝟐 because 𝒇 has a sharp turn at 𝒙=𝟐.
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§2.1A: Alternate Definition of a Derivative

24. §2.1A: Alternate Definition of a Derivative
Example 8
Use the graph to determine the derivative of 𝒇 𝒙 = 𝒙 𝟏/𝟑 at 𝒙 =𝟎. If it does not exist, explain why.
𝒇 is not differentiable at 𝒙=𝟐 because 𝒇 has a sharp turn at 𝒙=𝟐.
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§2.1A: Alternate Definition of a Derivative

25. §2.1A: Alternate Definition of a Derivative
Example 9
Use the graph to determine the derivative of 𝒇 𝒙 = 𝒙−𝟏 𝟐/𝟑 . If it does not exist, explain why.
𝒇 is not differentiable at 𝒙=𝟏
because 𝒇 has a cusp at 𝒙=𝟏.
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§2.1A: Alternate Definition of a Derivative

26. AP Multiple Choice Practice Question 1 (non-calculator)
Let 𝒇 be a function such that 𝐥𝐢𝐦 𝒉→𝟎 𝒇 𝟐+𝒉 −𝒇 𝟐 𝒉 =𝟓. Which of the following must be true?
𝒇 is continuous at 𝒙=𝟐
𝒇 is differentiable at 𝒙=𝟐
The derivative of 𝒇 is continuous at 𝒙=𝟐
(A) I only
(B) II only
(C) I and II only
(D) II and III only
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§2.1A: Alternate Definition of a Derivative

27. AP Multiple Choice Practice Question 1 (non-calculator)
Let 𝒇 be a function such that 𝐥𝐢𝐦 𝒉→𝟎 𝒇 𝟐+𝒉 −𝒇 𝟐 𝒉 =𝟓. Which of the following must be true?
Vocabulary
Connections and Process
Answer and Justifications
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§2.1A: Alternate Definition of a Derivative

28. §2.1A: Alternate Definition of a Derivative
Assignment
Worksheet
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§2.1A: Alternate Definition of a Derivative