Find the derivative of implicitly defined functions
I can recognize when implicit differentiation is needed to find a derivative
I can calculate derivatives of implicitly defined functions
I can calculate second derivatives
Quick Lesson Plan
In this lesson, the chain rule is applied to a new type of function --- implicitly defined functions --- as students work with the familiar equation for a circle centered at the origin and then find the derivative of each term. This first example asks students to recall the derivative of y in explicitly defined functions and to interpret the label dy/dx. The chain rule makes then makes its appearance when deriving the “inner” function of the y-squared term.
Take time to review the utility of explicitly and implicitly defined functions: because some equations cannot be solved for y, implicit definitions (and implicit differentiation) are the only option.
Stress the need to identify chain rule derivatives on every variable term: the derivative of “x” terms produces dx/dx terms which equal one and are usually ignored, but the derivative of “y” terms produces dy/dx terms which must be included in the derivative expression.
To impress upon students the usefulness of implicit differentiation, they should find derivatives using two methods: solve for y (if possible) and find the derivative of the explicitly defined function, and then try implicit differentiation to get (hopefully!) the identical derivative.
For an excellent example of how implicit differentiation may appear on the AP Calculus Test, see 2015 AB #6 and the accompanying scoring guidelines and student samples.
Implicit differentiation is a necessary skill for both the AB and BC student.
If students have been practicing the chain rule on explicitly defined functions, such as y equals x-squared, they should be familiar with the appearance of a dx/dx term. When the derivative of y equals x-squared has been frequently written as (2x)(dx/dx), students will quickly become comfortable with the introduction of a (dy/dx) term in the midst of a longer derivative expression. Later in the course, as parametric, vector, and polar functions are introduced, the derivative expressions will be easier to develop if students have seen many types of differential expressions.
“Malgebra” rears its ugly head in this lesson as students struggle to factor out and isolate the dy/dx term. Tomorrow’s activity presents an entertaining opportunity for students to sharpen their algebra skills in a low-anxiety group setting.