Implicitly Defined Functions

1. Implicitly Defined Functions Chapter 4.2

2. Implicitly Defined Functions • Any equation in and that can be solved for is said to be explicitly defined • The slope-intercept form of a linear equation is explicitly defined: • The general form of a linear equation is implicitly defined: • To say that it is defined implicitly is to say that is a function of , even though it isn’t solved for • It is a simple matter to convert the general form to the slope-intercept form, but this is not true for all equations in and

3. Implicitly Defined Functions • For example, the curve defined by is defined implicitly, and it cannot be solved for as a function of • By the vertical line test, this equation does not define a function • But we can think of this as the union of three separate functions, so it is proper for us to speak of as a function of • In fact, it is differentiable at all but two points • The problem is how to find the derivative if we cannot solve for

4. Implicitly Defined Functions

5. Implicitly Defined Functions • Implicit differentiation is the process by which we can find even if we cannot solve for • The process uses the Chain Rule • It is important to remember that we think of as a function of ! • We might rewrite the equation as • We see that is an “inner” function and we must treat it as such

6. Example 1: Differentiating Implicitly Find if .

7. Example 1: Differentiating Implicitly Find if . If we take , we can rewrite the equation as . Now we apply the Chain Rule on the left and differentiate normally on the right: Now solve for to get

8. Example 1: Differentiating Implicitly Find if . A simpler way is to use :

9. Example 1: Differentiating Implicitly Find if . We can also use Leibniz notation: It is common and acceptable to have a derivative expressed as a function of .

10. Example 2: Finding Slope on a Circle Find the slope of the circle at the point .

11. Example 2: Finding Slope on a Circle Find the slope of the circle at the point . Differentiate implicitly: The slope at is

12. Example 3: Solving for Show that the slope is defined at every point on the graph of .

13. Example 3: Solving for Show that the slope is defined at every point on the graph of . Note that we cannot solve this for . Remember that is an inner function of :

14. Example 3: Solving for Show that the slope is defined at every point on the graph of . The equation is differentiable so long as . But since the maximum and minimum values for cosine are , then the denominator cannot equal zero and the derivative is defined for all values of

15. Implicit Differentiation Process • Differentiate both sides of the equation with respect to • Collect the terms with on one side of the equation • Factor out • Solve for

16. Example 4: Tangent and Normal to an Ellipse Find the tangent and normal to the ellipse at the point .

17. Example 4: Tangent and Normal to an Ellipse Find the tangent and normal to the ellipse at the point .

18. Example 4: Tangent and Normal to an Ellipse Find the tangent and normal to the ellipse at the point . The slope of the tangent line at is The slope of the normal line is, thus, The tangent line equation is ; the normal line is

19. Example 5: Finding a Second Derivative Implicitly Find if .

20. Example 5: Finding a Second Derivative Implicitly Find if . First, find (or ): Now find the derivative of : Since we know , we can substitute it to get

21. Exercise 4.2