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