S ECTION 3.5 R ECAP Implicit Differentiation. U P TO THIS POINT...

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SECTION 3.5 RECAP Implicit Differentiation

Transcript of S ECTION 3.5 R ECAP Implicit Differentiation. U P TO THIS POINT...

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SECTION 3.5 RECAP Implicit Differentiation

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UP TO THIS POINT . . .

We’ve been deriving equations easily written explicitly as a function of .

Ex’s:

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BUT . . .

Some functions, however, are only implied by an equation.

What do we do in that case?

Ex:

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IMPLICIT DIFFERENTIATION

So, what does mean?

Thus, implicit differentiation . . .

is used when we aren’t given a function written nicely in terms of a dependent variable.

is accomplished by . . . differentiating with respect to the independent variable

in the usual way. differentiating the dependent variable using the chain

rule (i.e. like we did with the “”s and “”s)

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EXAMPLES

a. Explicit Example

b. Implicit Examples

Variables agree

Variables disagree

un u nun – 1

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DON’T FORGET THE

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GRAPHS OF FUNCTIONS AND DERIVATIVES

𝒚=𝟏𝟑𝒙𝟑−𝟒 𝒙

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GRAPHS OF FUNCTIONS AND DERIVATIVES

𝒚=𝟏𝟑𝒙𝟑−𝟒 𝒙

𝒚 ′=𝒙𝟐−𝟒

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GRAPHS OF FUNCTIONS AND DERIVATIVES

𝒚 ′=𝒙𝟐−𝟒

𝒚 ′ ′=𝟐 𝒙

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GRAPHS OF FUNCTIONS AND DERIVATIVES

𝒚=𝟏𝟑𝒙𝟑−𝟒 𝒙

𝒚 ′=𝒙𝟐−𝟒

𝒚 ′ ′=𝟐 𝒙

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SECTION 3.6Derivatives of Inverse Functions

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THINK BACK TO INVERSE FUNCTIONS . . .

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IS THERE A RELATIONSHIP BETWEEN THEIR DERIVATIVES?

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TWO THEOREMS

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AN EXAMPLE . . .

𝑓 (𝑥 )=𝑥3

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OUR MAIN FOCUS

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EXAMPLE 1Find an equation of the tangent line to the graph at the given point.

Equation Point

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EXAMPLE 2Find at the given point for the given equation.

Equation Point

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EXAMPLE 3Find the derivative of the function.a.

b.

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EXAMPLE 4Find the derivative of the function.

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EXAMPLE 5Find the derivative of the function.

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EXAMPLE 6Find an equation of the tangent line to the graph at the given point.

Equation Point

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WHAT WE’VE LEARNED THUS FAR