Implicit differentiation

In mathematics, an implicit function is a function in which the dependent variable has not been given explicitly in terms of the independent variable. Generally an explicit function determines the output value of the function y in terms of the input value x:
y = f(x).

But in an implicit function the value of y is obtained from x by solving an equation of the form:
R(x,y) = 0.

That is, it is defined as the level set of a function in two variables: one variable or the other may determine the other, but one is not given an explicit formula for one in terms of the other.

Implicit functions can often be useful in situations where it is inconvenient to solve explicitly an equation of the form R(x,y) = 0 for y in terms of x. Even if it is possible to rearrange this equation to obtain y as an explicit function f(x), it may not be desirable to do so since the expression of f may be much more complicated than the expression of R. In other situations, the equation R(x,y) = 0 may fail to define a function at all, and rather defines a kind of multiple-valued function. Nevertheless, in many situations, it is still possible to work with implicit functions. Some techniques from calculus, such as differentiation, can be performed with relative ease using implicit differentiation.

Definitions and examples

Implicit differentiation is nothing more than a special case of the well-known chain rule for derivatives. A familiar example of this is the equation

X² + y² = 25 ,
which represents a circle of radius five centered at the origin.

In calculus, implicit differentiation can be applied to implicit functions. This is by an application of the chain rule, to calculate derivatives without necessarily making y an explicit function of x. So, implicit differentiation is nothing more than a special case of the chain rule for derivatives.

Sometimes standard explicit differentiation cannot be used, and in order to obtain the derivative, another method such as implicit differentiation must be employed. An example of such a case is the implicit function y3 - y = x. It is impossible to express y explicitly as a function of x (at least using elementary means, although the cubic formula will suffice for restricted values of x and y), which means that cannot be found by explicit differentiation.

When we implicitly differentiate, we must treat y as a composite function and therefore we must use the chain rule with y terms. Implicit differentiation is a method of finding the derivative of an implicit function by taking the derivative of each term with respect to the independent variable while keeping the derivative of the dependent variable with respect to the independent variable in symbolic form and then solving for that derivative.