A function defined by an equation of the form [in general,
]. If
is thought of as the dependent variable,
is said to define
as an implicit
function of
.
Let y be related to x by the equation
and suppose the locus is that shown in figure.
We cannot say that is a function of
since at a particular value of
there is more than one
value of
(because, in the figure, a line perpendicular to the
axis intersects the locus at more than
one point) and a function is, by definition, single-valued. Although equation above does not define
as a
function of
, we can say that on certain judiciously chosen segments of the locus y can be considered to be a
single-valued function of
[expressible as
]. For example, the segment P1P2 could be separated
out as defining a function
. As a consequence, it is customary to say that equation defines
implicitly as a function of
; and we refer to
as an implicit function of
.
The implicit function theorem provides a condition under which a relation defines an implicit function. It states that
if the left hand side of the equation is differentiable and satisfies some mild condition on its
partial derivatives at some point
such that
, then it defines a function
over some interval containing
. Geometrically, the graph defined by
will overlap locally with
the graph of some equation
.
Consider the locus of shown in the figure. Let us ask the following question: “At a particular point
on the locus what is the value of the quantity
?” This question can be answered at all points on the locus
except points P1, P2, P3 and P4 (at these points the quantity
does not exist – it becomes infinite) and the
answer is:
If we have an equation of the type , and certain conditions are met, we can view one of the variables
as a function of the other in the vicinity of a particular point
that satisfies the equation. The
conditions that must be met are stated in the implicit function theorem.
If we have an equation such as which defines a variable as a function of others implicitly, there
are two techniques for computing derivatives.
Given a particular variable to be considered as the dependent variable, if it is possible to solve the equation for the dependent variable in terms of the independent variables, we can compute the derivative directly by formula.
Example. Compute for the equation
.
Solution. Solve the equation for to get
and compute the derivative directly as .
Decide which variable is to be considered the dependent variable and which the independent. Say y is to be considered
the dependent variable in . Regarding
as the dependent variable, differentiate the equation as it
stands with respect to the independent variable
and then solve the resulting relation for
.
This method is known as implicit differentiation.
Example. Compute for the equation
.
Solution. Differentiating implicitly, we get
Solving this for gives
Because in most cases it is difficult or impossible to solve for the dependent variable, we usually use the method of implicit differentiation.