Previous examples of derivatives were always set up in the form of:
This is a defined as y is an explicit function of x. There exist scenarios where y does not follow the rules of a function (a unique answer y for each value x) like the equation of a unit circle shown below.
As shown here, the unit circle equation is made up from two functions that are both valid and are said to be defined implicitly by the unit circle equation. To find the derivative of these types of equations, treat the y components as a function of y and solve the derivatives of each side of the function before rearranging as shown below.
There may be instances where the functions are a product of each other. The standard rules apply in these cases as shown below.