3.6 - Implicit 3.6 - Implicit DifferentiationDifferentiation

(page 211-217)(page 211-217)

We have been differentiating functions We have been differentiating functions that are expressed in the form that are expressed in the form y=f(x).y=f(x).

An equation in this form is said to define An equation in this form is said to define yy explicitly explicitly as a function of as a function of x.x.

The equations on the next slide are not The equations on the next slide are not defined defined explicitly explicitly although, the first although, the first two may be manipulated to be written two may be manipulated to be written as functions in the form as functions in the form y=f(x)y=f(x) the the second two equation are not functions.second two equation are not functions.

Implicit EquationsImplicit Equations

1. 1yx y x

2 23. 1x y

2. 1xy

3 34. 3x y xy

1

1

xy f x

x

1

y f xx

21y x Can not be written in terms of .y

Implicit EquationsImplicit Equations(page 212-213)(page 212-213)

In the original equations 1,2,3 on the In the original equations 1,2,3 on the previous slide we could say that the previous slide we could say that the equation defines equation defines yy implicitlyimplicitly as a as a function(s) of function(s) of x.x.

The 4th equation, called the“The 4th equation, called the“Folium of DescartesFolium of Descartes”, could not be written ”, could not be written explicitly to define explicitly to define y y in terms of in terms of x.x.

It is not necessary to solve an equation for It is not necessary to solve an equation for y y in terms of in terms of x x in order to differentiate the in order to differentiate the function defined implicitly by the equation.function defined implicitly by the equation.

Implicit DifferentiationImplicit Differentiation

We differentiate both sides of the We differentiate both sides of the equation and treat equation and treat y y as a composite as a composite function and apply the chain rule when function and apply the chain rule when necessary.necessary.

This method of obtaining derivatives is This method of obtaining derivatives is called called implicit differentiation.implicit differentiation.

We will use this technique to solve word We will use this technique to solve word problems like the one on the next slide problems like the one on the next slide involving involving “Related Rates”“Related Rates”

Related Rate Problem Related Rate Problem (page 220)(page 220)

Example 1Example 1(page 213)(page 213)

Example 1Example 1(page 248)(page 248)

Example 2Example 2(page 214)(page 214)

Example 3Example 3(page 214)(page 214)

Example 3Example 3(page 214)(page 214)

Example 3Example 3(page 214)(page 214)

Example 4Example 4(page 215)(page 215)

Example 4Example 4(page 215)(page 215)

Example 4Example 4(page 215)(page 215)

See graph on next slide.

Example 4Example 4(page 215)(page 215)

Example 4Example 4(page 215)(page 215)

See graph on next slide

Example 4Example 4(page 215)(page 215)

Differentiability of Differentiability of Functions Defined Functions Defined

ImplicitlyImplicitly(page 216)(page 216)

When differentiating implicitly, it is assumed When differentiating implicitly, it is assumed that that y y represents a differentiable function of represents a differentiable function of x. x.

If function is not differentiable, then the If function is not differentiable, then the derivative will be meaningless.derivative will be meaningless.

We will not be determining whether or not an We will not be determining whether or not an implicitly defined function is differentiable. “We implicitly defined function is differentiable. “We will leave such matters for more advanced will leave such matters for more advanced courses.courses.