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Section 3.8 Implicit Differentiation
MAT 1234 Calculus I Section 3.8 Implicit Differentiation
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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.
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WebAssign Reminder Be sure to simplify your answers.
WebAssign accepts un-simplified answers.
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Preview Introduce a new form of extended power rule (notations changes) Introduce Implicit Differentiation
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Recall (Extended Power Rule)
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Recall (Extended Power Rule)
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Recall (Extended Power Rule)
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Recall (Extended Power Rule)
We now free the variable 𝑦, which we need for the next formula.
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Recall (Extended Power Rule)
If 𝑢 is a function in 𝑥, then If y is a function in x, then
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Recall (Extended Power Rule)
If 𝑢 is a function in 𝑥, then If 𝑦 is a function in 𝑥, then
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Example 0
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The Needs for Implicit Differentiation…
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Example 1 Find the slopes of the tangent line on the graph i.e. find y
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Example 1: Method I Make y as the subject of the equation: y x
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Example 1: Method I Make y as the subject of the equation: y x
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Example 1: Method I Make y as the subject of the equation: y x
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Example 1: Method I Suppose the point is on the upper half circle y x
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Example 1: Method I Suppose the point is on the lower half circle y x
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Example 1: Method I Two disadvantages of Method I: ???
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Example 1: Method II Implicit Differentiation:
Differentiate both sides of the equation.
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Expectation You are required to show the implicit differentiation step
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Notations 𝑑 𝑑𝑥 (…) is the derivative of (…). Do not confuse it with 𝑑𝑦 𝑑𝑥 which is the derivative of 𝑦.
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Example 2
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Example 3 Find the slope of the tangent line at
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Notations Correct or Incorrect because
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Example 4
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