1. Implicit Differentiation
Objective: To find derivatives of functions that we cannot solve for y.

2. Implicit Differentiation
It is not necessary to solve an equation for y in terms of x in order to differentiate the function defined implicitly by the equation (but often it is easier to do so).
Find dy/dx for Can we solve this for y?

3. Implicit Differentiation
It is not necessary to solve an equation for y in terms of x in order to differentiate the function defined implicitly by the equation.
For example, we can take the derivative of
with the quotient rule:

4. Implicit Differentiation
We can also take the derivative of the given function without solving for y by using a technique called implicit differentiation. We will use all of our previous rules and state the independent variable.

5. Example 2
Use implicit differentiation to find dy/dx if

6. Example 2
Use implicit differentiation to find dy/dx if

7. Example 2
Use implicit differentiation to find dy/dx if

8. Example 2
Use implicit differentiation to find dy/dx if

9. Example 3
Use implicit differentiation to find if

10. Example 3
Use implicit differentiation to find if

11. Example 3
Use implicit differentiation to find if

12. Example 3
Use implicit differentiation to find if

13. Example 3
Use implicit differentiation to find if

14. Example 3
Use implicit differentiation to find if

15. Example 3
Use implicit differentiation to find if

16. Example 4 Find the slopes of the tangent lines to the curve
at the points (2, -1) and (2, 1).

17. Example 4 Find the slopes of the tangent lines to the curve
at the points (2, -1) and (2, 1).
We know that the slope of the tangent line means the value of the derivative at the given points.

18. Example 4 Find the slopes of the tangent lines to the curve
at the points (2, -1) and (2, 1).
We know that the slope of the tangent line means the value of the derivative at the given points.

19. Example 4 Find the slopes of the tangent lines to the curve
at the points (2, -1) and (2, 1).
We know that the slope of the tangent line means the value of the derivative at the given points.

20. Example 5
Use implicit differentiation to find dy/dx for the equation

21. Homework
Page
1-23 odd
27 (just use implicit)