Up to this point, we have been discussing graphs of functions
. In this case, it is explicitly clear how the dependent
variable
depends on the independent variable
. However, sometimes we are faced with an implicit
relationship between and .
For example, consider the equation for the circle of radius
:
. Here we can no longer write
as a function of : for values of
between
and
, there are
two values of on the
circle. Nevertheless, when we look at the circle, we might
expect that the notion of a tangent line and hence a derivative still
make sense.
However, we can no longer expect the derivative
to depend only on
, but rather on
the point
on the circle in which we are interested.
Implicit differentiation is a process which
will clarify this for us.
Implicit Functions
In spite of the fact that the circle cannot be described as the
graph of a function, we can describe various parts of the circle as
the graphs of functions. For instance, the upper semicircle is the
graph of
(this is an
implicit function defined by the equation). For the time being,
let's forget that we can explicitly solve for
. In
terms of the original equation of the circle, we have
Now we can differentiate this expression as a function of by applying the chain rule :
Now in terms of , we have
This final expression may appear dissatisfying to you since the
derivative of is expressed in terms of both and . However, this is a very useful
expression: if we know a point on the circle
, then
we know that the slope of the tangent line there is
. For instance, at the point
, we know that
. This feels right since, from the picture
above, we expect the
slope of the tangent line to be 1 at that point.
Notice something else important: we did not need to know
explicitly what
was. Instead, we only needed to
know that there was such a function. Seen in this light, the
computation we have done still holds for the bottom half of the circle
which may be described as the graph of
. For instance, at the point
we compute
that
which again agrees with our intuition.
Notice that the derivative is not defined when
. This again makes sense because, from the picture above, the
tangent line becomes vertical and then has a slope which is not
defined.
This process is known as implicit differentiation.

The power rule
Now let's have a look at another example:
. This is actually given to us in an explicit
waythat is, we know exactly how depends on . Nevertheless, we have not yet seen how to differentiate
this type of function.
However, we can rewrite this implicitly as
and implicitly differentiate as follows:
Notice that this result
still looks like the
power rule which we have seen for integer exponents. In fact, we
can use implicit differentiation to verify the power rule for any
fractional exponenent. This means that we can now write
whenever
is a rational number (i.e. a quotient of
two integers).

Another example
Here we will consider the implicit relationship
This is a famous curve called the astroid and it arises
in an interesting way. Suppose that a ball of radius
is rolling inside a ball of radius 2. The astroid is the curve
traced out by a point on the inner circle. You can see this below by
either dragging the red ball around or by starting the animation using
the "Start" button.
We can use implicit differentiation to understand its tangent
lines:
First of all, notice that if either
or
, then the derivative cannot be defined. You can see this by
looking at the animation above. When the point on the inner circle
comes to one of these points, it comes to a stop and changes its
direction. At these points, there is no tangent line and hence the
derivative cannot be defined.
At the point
,
however, we find that
.

For your consideration:
