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Section 3.8 Implicit Differentiation

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

2 Next… 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.

3 WebAssign Reminder Be sure to simplify your answers.
WebAssign accepts un-simplified answers.

4 Preview 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


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