AP Calculus Review Implicit Variation

Implicit variation (or implicit differentiation) is a powerful technique for finding derivatives of certain equations. In this review article, well see how to use the method of implicit variation on AP Calculus problems. What is Implicit Variation? The usual differentiation rules, such as power rule, chain rule, and the others, apply only to functions of the form y = f(x). In other words, you have to start with a function f that is written only in terms of the variable x. But what if you want to know the slope at a point on a circle whose equation is x2 + y2 = 16, for example? Circle of radius )(x 2) 3. (c) Implicit Second Derivatives To find the second derivative of an implicit function, you must take a derivative of the first derivative (of course!). However, all of the same peculiar rules about expressions of y still apply. Note that we are using the Quotient Rule to start things off. Now, the good news is that we dont have to simplify the expression any further. This is because they are looking for a numerical final answer. So we just have to plug in the given (x, y) coordinates. But what about the two spots where dy/dx shows up? Well we already have an expression for dy/dx from part (a). Simply plug in your (x, y) coordinates to find dy/dx and now you can plug that into the second derivative expression as well. Summary On the AP Calculus AB or BC exam, you will need to know the following. How to find the derivative of an implicitly-defined function using the Method of implicit variation (a.k.a. implicit differentiation). What the derivative means in terms of slope and how to find tangent lines to a curve defined implicitly. How to compute second derivatives of implicitly-defined functions.