So far, we have been concerned with differentiating functions that are expressed in the form y = f (x). We say that an equation of this form defines y explicitly as a function of x, because the variable y appears alone on one side of the equation. However, sometimes functions are defined with equations in which y It is not alone on one side; for example the equation
yx + y +1 = x
it's not in the form y = f (x). However, this equation still defines y as a function of xonce you can rewrite how
So we say that xy + y +1 = x define y implicitly as a function of x, being
f (x) =
An equation in x and y can implicitly define more than one function of x; for example if we solve the equation
for y in terms of xwe get ; thus we find two functions that are implicitly defined by , this is
The graphs of these functions are upper and lower semicircles of the circle. .
y = -
In general, if we have an equation in x and y, so any segment of your graph that passes the vertical test can be viewed as a graph of a function defined by the equation. Thus we make the following definition:
|Definition. We say that a given equation in x and y define the function f implicitly if the graph of y = f (x) match some segment of the equation graph.|
So, for example, the equation define the functions and implicitly, since the graphs of these functions are the segments of the circle .
Sometimes it can be difficult or impossible to solve an equation in x and y for y in terms of x.
With persistence, the equation
for example it can be resolved to y in terms of x, but algebra is boring and the resulting formulas are complicated. On the other hand, the equation
sen (xy) = y
cannot be resolved to y in terms of x by any elementary method. So even if an equation in x and y can define one or more functions of x, it may not be practical or possible to find explicit formulas for those functions.Next: Implicit Differentiation