# Implicit differentiation.

Sometimes a graph in the plane is defined by an equation of the form , but this equation cannot be put into the form . In such a case we cannot compute a derivative by the ordinary methods. Nevertheless, on some well-chosen intervals, it is possible to do it by implicit differentiation. We will see that on examples.

Example 6.15.1   Let be the ellipse whose equation is (displayed on Figure 17. At the vertices and , it has tangents parallel to the axis, therefore these tangents have no slope. But at any other point the tangent is well-defined; the reason is the following: the graph can be written as , where is the upper half ellipse (whose equation is and is the lower half ellipse (whose equation is . For , the function is differentiable and, at the corresponding point on the graph, there is a tangent which has a well-defined slope.

We will not do it here in this way. Consider the equation of :

We differentiate each side with respect to :

If , we have:

E.g. the slope of the tangent to at the point whose coordinates are is equal to .

Example 6.15.2   Let be the curve with equation . By implicit differentiation with respect to we have:

At every point such that , is a differentiable fuction of and we have:

E.g., the slope of the tangent to at the point whose coordinates are is equal to 0.

Noah Dana-Picard 2007-12-28