Some curves are described by equations of the form . For example, is the equation of circle of radius 1. For the upper semi-circle we can solved it for and write and for the lower semi-circle . There is no function near that satisfies .
Let . Suppose and has continuous first partial derivatives and so is differentiable. There are two questions that we try to answer:
Can we solve for as a function of near ? In other words, can we find a function defined on some interval (for ) such that
If such a function exists, what is
Suppose such a function exists. To find
Method (a): We use the chain rule:
so that
Method (b): Because is constant and therefore its the total differential is zero. If we divide the above equation by and assume , we obtain the same result as method (a).
We assumed that a function existed and then showed the condition was required for calculation of . In fact, it can be proved this condition, , is sufficient for the existence of with the aforementioned conditions. The condition means that the tangent line to the level curve is not vertical, and therefore, a part of the level curve — close enough to the point — can be the graph of the function . When , you may not be able to find such a function. For example in Fig. 1, if we just keep the shaded disk around and remove the rest of the level curve, what we get can be the graph of a function, because any vertical line now does not intersect this part of the curve more than once. However, at or , where the tangent line is vertical, we cannot find a disk around them (to keep the curve and remove the rest) where a vertical line does not intersect the level curve twice. Therefore the level curve near or cannot be the graph of a function.
Figure 1.
Noting and are both functions of and , higher derivatives of with respect to can be found by successive differentiation with respect to provided higher partial derivatives of exist:
Example 1
Given , find .
Solution
Let . Thus: and it follows that We cannot use the above formula when . As you can see in Fig. 2, when and , the tangent line to the curve is vertical.
Figure 2.
Now suppose the equation is given, where has continuous partial derivatives. If and , near can be written as a function of and , namely . In other words, the level surface can be locally the graph of a function . To find the partial derivatives of , we differentiate the equation with respect to and :
Therefore:
and
Example 2
Given , find and .
Solution
Let . Then
Therefore:
Note that and the above expressions are valid everywhere because for any point on the level surface, (note ).
In general we have the following theorem.
Theorem 1. (Implicit Function Theorem) Suppose and is of class (i.e., has continuous first partial derivatives). Assume:
where and . Then there is a neighborhood of in , a neighborhood of in , and a function of class such that if and satisfy , then . The partial derivatives of are given by: