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

2. Explicit vs. Implicit
An equation of the form y = f(x) is said to define y explicitly as a function of x because the variable y appears alone on one side of the equation.

3. Explicit vs. Implicit
An equation of the form y = f(x) is said to define y explicitly as a function of x because the variable y appears alone on one side of the equation.
Sometimes functions are defined by equations in which y is not alone on one side. For example
is not of the form y = f(x), but it still defines y as a function of x since it can be rewritten as

4. Explicit vs. Implicit We say that the first form of the equation
defines y implicitly as a function of x.

5. Explicit vs. Implicit
An equation in x and y can implicitly define more than one function in x. This can occur when the graph of the equation fails the vertical line test, so it is not the graph of a function.

6. Explicit vs. Implicit
An equation in x and y can implicitly define more than one function in x. This can occur when the graph of the equation fails the vertical line test, so it is not the graph of a function.
For example, if we solve the equation of the circle
for y in terms of x, we obtain
This gives us two functions that are defined implicitly.

7. Explicit vs. Implicit Definition 3.1.1
We will say that a given equation in x and y defines the function f implicitly if the graph of y = f(x) coincides with a portion of the graph of the equation.

8. 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.

9. 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:

10. Implicit Differentiation
We can also take the derivative of the given function without solving for x by using a technique called implicit differentiation.

11. 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.

12. 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.

13. 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.

14. 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.

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

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

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

18. Example 3
Use implicit differentiation to find if

19. Example 3
Use implicit differentiation to find if

20. Example 3
Use implicit differentiation to find if

21. Example 3
Use implicit differentiation to find if

22. Example 3
Use implicit differentiation to find if

23. Example 3
Use implicit differentiation to find if

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

25. 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.

26. 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.

27. 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.

28. Example 5
Use implicit differentiation to find dy/dx for the equation
Find the equation of the tangent line at the point

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

30. Example 5
Use implicit differentiation to find dy/dx for the equation
Find the equation of the tangent line at the point

31. Homework
Section 3.1
Page 190
1-19 odd
19 (just use implicit)