4-A Implicit Differentiation Examples
Instructions
The applet includes 5 examples of graphs of implicit equations. Use the slider tool to change the example. Check the "Normal Line" box to show/hid the normal line (i.e., the line perpendicular to the tangent line).
4-A Implicit Differentiation
We will come across some situations where we can describe a relationship between two variables with an equation, but we can't solve the equation to isolate one variable as the output of a function formula. Graphically, this typically means that the graph of the equation does not pass the vertical line test, i.e., there can be multiple y-values paired with a particular x-value. Under certain conditions we can say that one of the variables is an implicit function of the other. This means that we can't find an explicit function formula, but that if we look at a small enough section of the graph (called a branch) we would be able to treat the equation more like a function.
When working with these equations it still makes sense to talk about a tangent line on the graph at each point and therefore we can still think of the derivative as the slope of a tangent line at a point on the graph. We will just have to introduce a new technique, called implicit differentiation, in order to find these derivatives. Essentially this technique will rely on the Chain Rule, treating any occurrence of y as a function of x.