Implicit Differentiation AP Calculus October 26 27 2016

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

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

Explicit Functions n n 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. q y is written explicitly in terms of x.

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

Explicit Functions n n 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.

Implicit Relations n n n 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.

Implicit Differentiation 1. 2. 3. Differentiate both sides of the equation with respect to x (putting as derivative of y). Move all terms with the equation. dy/ dx to one side of 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)

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

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

Orthogonal Trajectories n n 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

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