1. Implicit Differentiation
AP Calculus
October 26-27, 2016
Mrs. Agnew

2. Essential Question Essential Vocabulary
How do you find the derivatives of implicit relations?
Essential Vocabulary
Explicit Function
Implicit Relation
Implicit Differentiation

3. Explicit Functions
Thus far, all differentiable functions we have worked with have had y written in terms of x.
When a function can be written as “y=”, it is called an EXPLICIT FUNCTION.
y is written explicitly in terms of x.

4. Explicit Functions
Finding derivatives of explicit functions requires the use of the differentiation rules discussed…

5. Explicit Functions
Notice that derivatives of explicit functions ask for the derivative of y in terms of x…
Finding of explicit functions is easy due to the fact that y is written in terms of x.

6. Implicit Relations
An implicit equation is an equation in which y cannot be written in terms of x.
For example…
We will have to use an alternate method to find the derivative of y in terms of x… IMPLICIT DIFFERENTIATION.

7. Implicit Differentiation
Differentiate both sides of the equation with respect to x (putting dy/dx as derivative of y).
Move all terms with dy/dx to one side of the equation.
Factor out dy/dx and solve for it by dividing both sides by resulting factor.
Stewart Examples: pg 243 #2 – 8 (E), 12 – 18 (E)

8. Homework: 10/26/2016
page 146 – 148
#5, 7, 9, 11, 15, 24
26, 29, 33, 37, 54, 57

9. Second Derivatives
It is possible to find the second derivative of an implicit equation by using implicit differentiation twice.
Replace dy/dx in second derivative with first derivative
Simplify expression
Examples (#35, 37)

10. Orthogonal Trajectories
Two curves are orthogonal if at each point of intersection their tangent lines are perpendicular.
To show that two curves are orthogonal trajectories, find derivative of each and show that they are opposite reciprocals.
Examples

11. Homework: 10/27/16
page 146 – 148
#8, 10, 23, 25, 31, 32, 34,
39, 47, 49, 53, 58, 59, 61