Implicit Equations are often used to represent relations that cannot be expressed as functions (one to many relations or many to many relations). These often have really interesting graphs. This page will allow to explore some of these graphs and consider why we need to use implicit diffrentiation. If you are looking for help with the techniques of implicit differentiation, examples and exam-style questions then you should visit Implicit Differentiation .
Here are the graphs of some implicit relations
A circle is the locus of a point that is a fixed distance from a point
The equation can be written implicity like this \(x^2+y^2=a^2\) where a is the radius of the circle. Drag the point around the circle to consider the gradient of this relation.
We can write the eqution of this circle explicity \(y=\pm\sqrt{a^{2}-x^{2}}\)
This creates 2 derivatives: one for when y>0 and one for when y<0.
Let’s take the case when the radius if the circle is equal to 1. The graph of \(y=\pm\sqrt{1^{2}-x^{2}} \)for y>0 is plotted below.
Sketch the graph of the gradient function. You can check your answer by clicking in the box.
You can find the gradient function of this explicit function using the chain rule. Try it for yourself!
Answer
How much of Graphs of Implicit Equations have you understood?