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 *x*once you can rewrite how

* y* =

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 *x*we get ; thus we find two functions that are implicitly defined by , this is

and

The graphs of these functions are upper and lower semicircles of the circle. .

*y =*

*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.