1. Chapter : Derivatives Section 3.7: Implicit Differentiation
AP CALCULUS AB
Chapter :
Derivatives
Section 3.7:
Implicit Differentiation

2. What you’ll learn about
Implicitly Defined Functions
Lenses, Tangents, and Normal Lines
Derivatives of Higher Order
Rational Powers of Differentiable Functions
… and why
Implicit differentiation allows us to find derivatives of functions that are not defined or written explicitly as a function of a single variable.

3. Implicitly Defined Functions

4. Section 3.7 – Implicit Differentiation
Explicit Form of an equation – is expressed in the form
Ex:
Implicit Form of an equation – when both x and y are used throughout the equation

5. Implicitly Defined Functions

6. Example Implicitly Defined Functions

7. Implicit Differentiation Process

8. Section 3.7 – Implicit Differentiation
Guidelines for Implicit Differentiation:
1. Differentiate both sides of the equation with respect to x. (Treat y as a function of x)
2. Collect all terms involving on the left side of
the equation, and move all other terms to the right side of the equation.
3. Factor out of the left side of the equation.
4. Solve for by dividing the right side of the
equation by the part of the left side of the equation without the

9. Section 3.7 – Implicit Differentiation
Example:

10. Section 3.7 – Implicit Differentiation
Another example:

11. 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.

12. Lenses, Tangents and Normal Lines

13. Example Lenses, Tangents and Normal Lines

14. Example Lenses, Tangents and Normal Lines

15. Example Derivatives of a Higher Order

16. Section 3.7 – Implicit Differentiation
Higher Order Derivatives:

17. Section 3.7 – Implicit Differentiation

18. Rule 9 Power Rule For Rational Powers of x