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 y2:
the derivative of xy2:
and the derivative of 2:
Solving for we find that
Thus, whenever we differentiate an implicit function, the derivative will usually contain terms in both x and y.
Find the derivative of the following: