1. Implicit Differentiation
Section 2.5
Calculus AP/Dual, Revised ©2017
5/9/2019 4:30 AM
§2.5: Implicit Differentiation

2. §2.5: Implicit Differentiation
Review
Solve 𝒚′ or 𝒅 𝒅𝒙 of 𝒙𝟒+𝟑𝒚=𝟔𝒙
5/9/2019 4:30 AM
§2.5: Implicit Differentiation

3. §2.5: Implicit Differentiation
Review
This form is written in explicit form, as it means it represents what 𝒚 equals (or where the variable is on one side)
5/9/2019 4:30 AM
§2.5: Implicit Differentiation

4. §2.5: Implicit Differentiation
Review
Solve 𝒅 𝒅𝒙 of 𝟕 𝒙 𝟐
5/9/2019 4:30 AM
§2.5: Implicit Differentiation

5. §2.5: Implicit Differentiation
General Practice
Take the derivative in respects to 𝒙 or known as 𝒅 𝒅𝒙 𝒙
2) 𝟐𝒙𝟑
3) 𝟒𝒙𝟓
4) 𝟐 𝐜𝐨𝐬 (𝒙)
5/9/2019 4:30 AM
§2.5: Implicit Differentiation

6. §2.5: Implicit Differentiation
General Practice
Take the derivative in respects to 𝒚 or known as 𝒅𝒚 𝒅𝒙 𝒚
2) 𝟐𝒚𝟑
3) 𝟒𝒚𝟓
4) 𝟐 𝐜𝐨𝐬 (𝒚)
5/9/2019 4:30 AM
§2.5: Implicit Differentiation

7. §2.5: Implicit Differentiation
Definitions
Implicit Differentiation takes the derivative with both sides with respects to 𝒙 in a relationship between 𝒙, 𝒚, and 𝒅𝒚 𝒅𝒙 ; essentially an application of the CHAIN RULE
An IMPLICIT function is written to which one has to complete additional steps to isolate the variable. Functions written are not solved for y as it is not clearly stated but it is implied.
Examples include:
𝒚= 𝒙 𝟐 whereas it can be written implicitly as 𝒙𝒚=𝟏
𝒙 𝟐 −𝟐 𝒚 𝟑 +𝟒𝒚=𝟐 can be written with implicit differentiation
5/9/2019 4:30 AM
§2.5: Implicit Differentiation

8. §2.5: Implicit Differentiation
Steps
Differentiate both sides of the equation with respect to 𝒙. MUST Include a 𝒅𝒚 𝒅𝒙 or 𝒚′ every time to differentiate with 𝒚
Collect all 𝒅𝒚 𝒅𝒙 on LEFT side and all other “stuff” to the right side
Factor out 𝒅𝒚 𝒅𝒙
Divide both sides by the extra “stuff”
5/9/2019 4:30 AM
§2.5: Implicit Differentiation

9. §2.5: Implicit Differentiation
Example 1
Solve 𝒅𝒚 𝒅𝒙 of 𝒙 𝟐 + 𝒚 𝟐 =𝟐𝟓
𝒅 𝒅𝒚 𝟐𝒚 𝒅𝒚 𝒅𝒙
Do we have any 𝒚 terms or any terms we took the derivative in respects to 𝒚?
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§2.5: Implicit Differentiation

10. §2.5: Implicit Differentiation
Example 1
Solve 𝒅𝒚 𝒅𝒙 of 𝒙 𝟐 + 𝒚 𝟐 =𝟐𝟓
5/9/2019 4:30 AM
§2.5: Implicit Differentiation

11. §2.5: Implicit Differentiation
Example 2
Solve 𝒅𝒚 𝒅𝒙 of 𝟐 𝒙 𝟑 =𝟐 𝒚 𝟐 +𝟓
5/9/2019 4:30 AM
§2.5: Implicit Differentiation

12. §2.5: Implicit Differentiation
Example 3
Solve 𝒅𝒚 𝒅𝒙 of 𝟑 𝒚 𝟑 + 𝒚 𝟐 −𝟐𝒚+𝟒 𝒙 𝟐 =𝟕
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§2.5: Implicit Differentiation

13. §2.5: Implicit Differentiation
Your Turn
Solve 𝒅𝒚 𝒅𝒙 of 𝒚 𝟑 + 𝒚 𝟐 −𝟓𝒚− 𝒙 𝟐 =−𝟒
5/9/2019 4:30 AM
§2.5: Implicit Differentiation

14. x3y3 is being multiplied: §2.5: Implicit Differentiation
Example 4
Solve 𝒅𝒚 𝒅𝒙 of 𝒙 𝟑 𝒚 𝟑 −𝒚=𝒙
x3y3 is being multiplied:
Use the PRODUCT RULE
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§2.5: Implicit Differentiation

15. §2.5: Implicit Differentiation
Example 4
Solve 𝒅𝒚 𝒅𝒙 of 𝒙 𝟑 𝒚 𝟑 −𝒚=𝒙
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§2.5: Implicit Differentiation

16. §2.5: Implicit Differentiation
Example 5
Solve 𝒅𝒚 𝒅𝒙 of 𝐬𝐢𝐧 𝒚 𝐜𝐨𝐬 𝒙=𝟓𝒙
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§2.5: Implicit Differentiation

17. §2.5: Implicit Differentiation
Example 6
Solve 𝒅𝒚 𝒅𝒙 of 𝐬𝐢𝐧 𝒙= 𝒆 𝒚 in 𝟎, 𝝅 in terms of 𝒙.
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§2.5: Implicit Differentiation

18. §2.5: Implicit Differentiation
Example 6
Solve 𝒅𝒚 𝒅𝒙 of 𝐬𝐢𝐧 𝒙= 𝒆 𝒚 in 𝟎, 𝝅 in terms of 𝒙.
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§2.5: Implicit Differentiation

19. §2.5: Implicit Differentiation
Your Turn
Solve 𝒅𝒚 𝒅𝒙 of 𝟐 𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒚=𝟏
5/9/2019 4:30 AM
§2.5: Implicit Differentiation

20. §2.5: Implicit Differentiation
Example 7
Solve the second derivative of 𝒙 𝟐 + 𝒚 𝟐 =𝟏𝟔
5/9/2019 4:30 AM
§2.5: Implicit Differentiation

21. §2.5: Implicit Differentiation
Example 7
Solve the second derivative of 𝒙 𝟐 + 𝒚 𝟐 =𝟏𝟔
5/9/2019 4:30 AM
§2.5: Implicit Differentiation

22. §2.5: Implicit Differentiation
Example 7
Solve the second derivative of 𝒙 𝟐 + 𝒚 𝟐 =𝟏𝟔
5/9/2019 4:30 AM
§2.5: Implicit Differentiation

23. §2.5: Implicit Differentiation
Example 7
Solve the second derivative of 𝒙 𝟐 + 𝒚 𝟐 =𝟏𝟔
5/9/2019 4:30 AM
§2.5: Implicit Differentiation

24. §2.5: Implicit Differentiation
Example 7
Solve the second derivative of 𝒙 𝟐 + 𝒚 𝟐 =𝟏𝟔
5/9/2019 4:30 AM
§2.5: Implicit Differentiation

25. §2.5: Implicit Differentiation
Example 8
Solve the equation of the tangent line to the graph of 𝒙 𝟐 + 𝒚 𝟐 =𝟐𝟓 at the point 𝟑, −𝟒 .
5/9/2019 4:30 AM
§2.5: Implicit Differentiation

26. §2.5: Implicit Differentiation
Example 8
Solve the equation of the tangent line to the graph of 𝒙 𝟐 + 𝒚 𝟐 =𝟐𝟓 at the point 𝟑, −𝟒 .
5/9/2019 4:30 AM
§2.5: Implicit Differentiation

27. §2.5: Implicit Differentiation
Example 8
Solve the equation of the tangent line to the graph of 𝒙 𝟐 + 𝒚 𝟐 =𝟐𝟓 at the point 𝟑, −𝟒 .
5/9/2019 4:30 AM
§2.5: Implicit Differentiation

28. §2.5: Implicit Differentiation
Example 9
Find the equation of the tangent line to the graph of 𝟕 𝒙 𝟐 +𝒙𝒚+𝟕 𝒚 𝟐 =𝟏𝟓 at the point 𝟏, 𝟏 .
5/9/2019 4:30 AM
§2.5: Implicit Differentiation

29. §2.5: Implicit Differentiation
Example 9
Find the equation of the tangent line to the graph of 𝟕 𝒙 𝟐 +𝒙𝒚+𝟕 𝒚 𝟐 =𝟏𝟓 at the point 𝟏, 𝟏 .
5/9/2019 4:30 AM
§2.5: Implicit Differentiation

30. §2.5: Implicit Differentiation
Your Turn
Solve the equation of the tangent line to the graph of 𝒙 𝟐 +𝟒 𝒚 𝟐 =𝟒 at the point 𝟐 ,− 𝟏 𝟐 .
5/9/2019 4:30 AM
§2.5: Implicit Differentiation

31. AP Multiple Choice Practice Question 1 (non-calculator)
If 𝟓 𝒙 𝟐 −𝟐𝒙𝒚+𝟕 𝒚 𝟐 =𝟎, then solve 𝒅𝒚 𝒅𝒙 (A) 𝟓𝒙+𝟕𝒚 𝒙 (B) 𝒚−𝟓𝒙 𝟕𝒚 (C) 𝒚−𝟓𝒙 𝟕𝒚−𝒙 (D) 𝟏𝟎𝒙−𝟐𝒚+𝟏𝟒𝒚
5/9/2019 4:30 AM
§2.5: Implicit Differentiation

32. AP Multiple Choice Practice Question 1 (non-calculator)
If 𝟓 𝒙 𝟐 −𝟐𝒙𝒚+𝟕 𝒚 𝟐 =𝟎, then solve 𝒅𝒚 𝒅𝒙
Vocabulary
Connections and Process
Answer and Justifications
5/9/2019 4:30 AM
§2.5: Implicit Differentiation

33. AP Multiple Choice Practice Question 2 (non-calculator)
If 𝐭𝐚𝐧 𝒙+𝒚 =𝒙, then 𝒅𝒚 𝒅𝒙 = (A) 𝐭𝐚𝐧 𝟐 𝒙+𝒚 (B) 𝐬𝐞𝐜 𝟐 𝒙+𝒚 (C) 𝐬𝐢𝐧 𝟐 𝒙+𝒚 −𝟏 (D) 𝐜𝐨𝐬 𝟐 𝒙+𝒚 −𝟏
5/9/2019 4:30 AM
§2.5: Implicit Differentiation

34. AP Multiple Choice Practice Question 2 (non-calculator)
If 𝐭𝐚𝐧 𝒙+𝒚 =𝒙, then 𝒅𝒚 𝒅𝒙 =
Vocabulary
Connections and Process
Answer and Justifications
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§2.5: Implicit Differentiation

35. §2.5: Implicit Differentiation
Assignment
Worksheet
5/9/2019 4:30 AM
§2.5: Implicit Differentiation