Editing Differential calculus (section)

=== Implicit function theorem ===
{{Main|Implicit function theorem}}

Some natural geometric shapes, such as [[circle]]s, cannot be drawn as the [[graph of a function]]. For instance, if {{math|''f''(''x'', ''y'') {{=}} ''x''<sup>2</sup> + ''y''<sup>2</sup> − 1}}, then the circle is the set of all pairs {{math|(''x'', ''y'')}} such that {{math|''f''(''x'', ''y'') {{=}} 0}}. This set is called the zero set of {{math|''f''}}, and is not the same as the graph of {{math|''f''}}, which is a [[paraboloid]]. The implicit function theorem converts relations such as {{math|''f''(''x'', ''y'') {{=}} 0}} into functions. It states that if {{math|''f''}} is [[continuously differentiable]], then around most points, the zero set of {{math|''f''}} looks like graphs of functions pasted together. The points where this is not true are determined by a condition on the derivative of {{math|''f''}}. The circle, for instance, can be pasted together from the graphs of the two functions {{math|± {{sqrt|1 - ''x''<sup>2</sup>}}}}. In a neighborhood of every point on the circle except {{nobreak|(−1, 0)}} and {{nobreak|(1, 0)}}, one of these two functions has a graph that looks like the circle. (These two functions also happen to meet {{nobreak|(−1, 0)}} and {{nobreak|(1, 0)}}, but this is not guaranteed by the implicit function theorem.)

The implicit function theorem is closely related to the [[inverse function theorem]], which states when a function looks like graphs of [[invertible function]]s pasted together.