3.7 Implicit Differentiation

xf(x)f(x)g(x)g(x) f ‘ (x)g ‘ (x) 2821/3 33 33 4422 5

xf(x)f(x)g(x)g(x) f ‘ (x)g ‘ (x) 2821/3 33 33 4422 5

In Exercises 1 –5, sketch the curve defined by the equation and find two functions y 1 and y 2 whose graphs will combine to give the curve. Quick Review

In Exercises 6 –8, solve for y ‘ in terms of y and x. Quick Review

In Exercises 9 and 10, find an expression for the function using rational powers rather than radicals. Quick Review

What you’ll learn about Implicitly Defined Functions Lenses, Tangents, and Normal Lines Derivatives of Higher Order Rational Powers of Differentiable Functions Essential Question How does implicit differentiation allows us to find derivatives of functions that are not defined or written explicitly as a function of a single variable?

Implicitly Defined Functions

Implicit Differentiation Process

Example Implicitly Defined Functions

Lenses, Tangents and Normal Lines In the law that describes how light changes direction as it enters a lens, the important angles are the angles the light makes with the line perpendicular to the surface of the lens at the point of entry (angles A and B in Figure 3.50). This line is called the normal to the surface at the point of entry. In a profile view of a lens, the normal is a line perpendicular to the tangent to the profile curve at the point of entry. Implicit differentiation is often used to find the tangents and normals of lenses described as quadratic curves.

Example Lenses, Tangents and Normal Lines 4.Find the lines that are (a) tangent and (b) normal to the curve at the given point.

Rule 9 Power Rule For Rational Powers of x

Pg. 162, 3.7 #2-42 even, 50, all

Find the x-value of the position(s) of the horizontal tangents of each equation. Review for Quiz

Find the x-value of the position(s) of the horizontal tangents of each equation. Review for Quiz

Find the x-value of the position(s) of the horizontal tangents of each equation. Review for Quiz