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

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Transcript of S ECTION 3.5 R ECAP Implicit Differentiation. U P TO THIS POINT...
SECTION 3.5 RECAP Implicit Differentiation
UP TO THIS POINT . . .
We’ve been deriving equations easily written explicitly as a function of .
Ex’s:
BUT . . .
Some functions, however, are only implied by an equation.
What do we do in that case?
Ex:
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)
EXAMPLES
a. Explicit Example
b. Implicit Examples
Variables agree
Variables disagree
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DON’T FORGET THE
GRAPHS OF FUNCTIONS AND DERIVATIVES
𝒚=𝟏𝟑𝒙𝟑−𝟒 𝒙
GRAPHS OF FUNCTIONS AND DERIVATIVES
𝒚=𝟏𝟑𝒙𝟑−𝟒 𝒙
𝒚 ′=𝒙𝟐−𝟒
GRAPHS OF FUNCTIONS AND DERIVATIVES
𝒚 ′=𝒙𝟐−𝟒
𝒚 ′ ′=𝟐 𝒙
GRAPHS OF FUNCTIONS AND DERIVATIVES
𝒚=𝟏𝟑𝒙𝟑−𝟒 𝒙
𝒚 ′=𝒙𝟐−𝟒
𝒚 ′ ′=𝟐 𝒙
SECTION 3.6Derivatives of Inverse Functions
THINK BACK TO INVERSE FUNCTIONS . . .
IS THERE A RELATIONSHIP BETWEEN THEIR DERIVATIVES?
TWO THEOREMS
AN EXAMPLE . . .
𝑓 (𝑥 )=𝑥3
OUR MAIN FOCUS
EXAMPLE 1Find an equation of the tangent line to the graph at the given point.
Equation Point
EXAMPLE 2Find at the given point for the given equation.
Equation Point
EXAMPLE 3Find the derivative of the function.a.
b.
EXAMPLE 4Find the derivative of the function.
EXAMPLE 5Find the derivative of the function.
EXAMPLE 6Find an equation of the tangent line to the graph at the given point.
Equation Point
WHAT WE’VE LEARNED THUS FAR