Lesson: Derivative Techniques - 4  Objective – Implicit Differentiation

 These functions are relatively easy to differentiate because they are defined “EXPLICITLY”. (meaning they are solved for y, in other words y is by itself on one side of the equation.)  Differentiate:

1.Implicit Differentiation – The process of finding the derivative of a function that is not solved for y.  Implicit functions do not have y isolated on one side of the equation.  To do this, you need to use the chain rule, with a little creativity mixed in.

EX 1: Differentiate  This function is not defined explicitly, but can be with the aid of a little algebra. Group all y terms on 1 side of the = sign. Factor out a y Divide by (x + 1) to isolate y

Now use the quotient rule To differentiate

 Question: O.K. but what about those that cannot be expressed explicitly or that are a nightmare to do so? I know you will be giving us some of those, won’t you, Mr. Winter! Well, let’s work our way into those. Let me show you first how to do the problem from example 1 “IMPLICITLY”.

Process of Implicit Differentiation Differentiate both sides of the equation with respect to x. Apply Chain Rule whenever you have an expression involving y. Move all terms involving dy/dx (or y’) to the left side of the equation, and everything else to the right side. Factor out dy/dx (or y’) on the left. Solve for dy/dx (or y’)

Now let’s try it. Differentiate:

But that answer doesn’t look the same as the One we got by defining it explicitly! Remember from the first method that: Substitute that in for y and See what happens.

EX. 2: Differentiate

EX. 3: Use implicit differentiation to find dy/dx if

EX. 4: Use implicit differentiation to find the second derivative d 2 y/dx 2 of  Solution:

EX. 5: Find the slopes of the tangent lines to the curve y 2 – x + 1 = 0 at the points (2, -1) and (2, 1).  Solution:

EX. 6: (a)Use implicit differentiation to find dy/dx for the “Folium of Descartes: x 3 + y 3 =3xy

 Divide all terms by 3 to simplify

EX. 7:

HW 4.1 Pg. 241(1 – 3, 5, 11 – 15, 21 – 24, 30*)