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).

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 [b]implicit function[/b] 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 [i]branch[/i]) we would be able to treat the equation more like a function. [br][br]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 [b]implicit differentiation[/b], in order to find these derivatives. Essentially this technique will rely on the [b]Chain Rule[/b], treating any occurrence of y as a function of x.