There are two main ways of describing a function, explicitly and implicitly .Defining a function explicitly, we have y=f(x).
A functional relation of the form f(x,y)=0 defines y as a function of x implicitly. In this case, we could try to solve for y as a function of x (from f(x,y)=0, a nonlinear equation!) and then, if this is successful, differentiate the resulting formula. What if we cannot find y as a function of x?
The way out of this dilemma is the Chain Rule...
For example, suppose y is defined via x 2 + y 2 = 1, it follows that
d/dx [ x 2 + y 2 ] = d/dx [ 1 ] = 0 Consequently,
2 x + 2 y(x) d/dx [ y(x) ] = 0 Therefore, dy/dx = - x/y(x).
Note: Given y as an implicit function of x, all derivatives of y will (in general) be implicit functions of x.
Note: To find whether a given point (x0,y0) is on the graph of an implicit function, it must be true that f(x0,y0)=0. Given a point x0, this necessitates solving a nonlinear equation f(x0,y)=0 for y.
Definition: Two families of curves are said to be orthogonal if the tangent lines are orthogonal at each point of intersection. An example is the set of hyperbolic curves xy=constant, and x2 - y2=constant. Such families of curves extend the notion of Cartesian coordinates in a natural way. These curves also result in electrostatics.