Figure N3–7We’ve inserted arrows to indicate the direction of motion.
Note that if your calculator can draw the curves in simultaneous graphing
mode, you can watch the objects as they move, seeing that they do indeed pass
through the intersection point at the same time.
G. IMPLICIT DIFFERENTIATION
When a functional relationship between x and y is defined by an equation of the
form F(x,y) = 0, we say that the equation defines y implicitly as a function of x.
Some examples are x^2 + y^2 − 9 = 0, y^2 − 4x = 0, and cos (xy) = y^2 − 5 (which can
be written as cos (xy) − y^2 + 5 = 0). Sometimes two (or more) explicit functions
are defined by F(x,y) = 0. For example, x^2 + y^2 − 9 = 0 defines the two functions
and , the upper and lower halves, respectively, of the
circle centered at the origin with radius 3. Each function is differentiable except
at the points where x = 3 and x = −3.
Implicit differentiation is the technique we use to find a derivative when y is
not defined explicitly in terms of x but is differentiable.
In the following examples, we differentiate both sides with respect to x, using
appropriate formulas, and then solve for .
Example 27 __
If x^2 + y^2 − 9 = 0, then