1. Section 3.8 Implicit Differentiation
MAT 1234 Calculus I
Section 3.8
Implicit Differentiation

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WebAssign 3.8
Quiz on 3.7, 3.8. Be sure to prepare for it. Remember, good study skills help you to do well in exams.

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Introduce a new form of extended power rule (notations changes)
Introduce Implicit Differentiation

5. Recall (Extended Power Rule)

6. Recall (Extended Power Rule)

7. Recall (Extended Power Rule)

8. Recall (Extended Power Rule)
We now free the variable 𝑦, which we need for the next formula.

9. Recall (Extended Power Rule)
If 𝑢 is a function in 𝑥, then
If y is a function in x, then

10. Recall (Extended Power Rule)
If 𝑢 is a function in 𝑥, then
If 𝑦 is a function in 𝑥, then

11. Example 0

12. The Needs for Implicit Differentiation…

13. Example 1 Find the slopes of the tangent line on the graph i.e. find y

14. Example 1: Method I
Make y as the subject of the equation: y x

15. Example 1: Method I
Make y as the subject of the equation: y x

16. Example 1: Method I
Make y as the subject of the equation: y x

17. Example 1: Method I
Suppose the point is on the upper half circle y x

18. Example 1: Method I
Suppose the point is on the lower half circle y x

19. Example 1: Method I
Two disadvantages of Method I:
???

20. Example 1: Method II Implicit Differentiation:
Differentiate both sides of the equation.

21. Expectation
You are required to show the implicit differentiation step

22. Notations
𝑑 𝑑𝑥 (…) is the derivative of (…). Do not confuse it with 𝑑𝑦 𝑑𝑥 which is the derivative of 𝑦.

23. Example 2

24. Example 3
Find the slope of the tangent line at

25. Notations
Correct or
Incorrect
because

26. Example 4