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Differentiation of Implicit and Inverse Functions

In the case of an implicit function, where

is treated as a function of

, and each term is differentiated with respect to

to find

For the equation of a circle,

is not explicitly a function of

. However:

When

In each case,

becomes a function of

defined over the closed interval

. Generally, for an equation

, by restricting the domain of

and

can become a function of

. In this context, the equation

is referred to as an implicit function of

in terms of

.
Additionally, the points

satisfying the equation

represent a curve on the coordinate plane.

Consider the equation

. By treating

as a function of

and differentiating each term with respect to

Thus,

as long as

If a differentiable function

has an inverse

, and if the inverse is differentiable, then: