IMPLICIT FUNCTIONS
In
equations containing x and y, separating the variables is not always easy. If
we do not solve an equation for y, we call y an implicit function of x. In the equation
y
is an implicit function of x, and x is also called an implicit function of y.
If we solve this equation for y, that is
then
y would be called an explicit function of x. In many cases such a solution
would be far too complicated to handle conveniently.
When
y is given by an equation such as
y
is an implicit function of x.
Whenever
we have an equation of this type in which y is an implicit function of x, we
can differentiate the function in a straightforward manner. The derivative of
each term containing y will be followed by
. Refer to Theorem 6.
EXAMPLE. Obtain the derivative
of
SOL
UTI0Y Find the derivative of y^{2}:
the
derivative of xy^{2}:
and
the derivative of 2:
such
that,
Solving
for
we find that
and
Thus,
whenever we differentiate an implicit function, the derivative will usually
contain terms in both x and y.
PRACTICE
PROBLEMS:
Find
the derivative
of the following:
ANSWERS:
