Editing Inverse function (section)

===Inverses and derivatives===
By the [[inverse function theorem]], a [[continuous function]] of a single variable <math>f\colon A\to\mathbb{R}</math> (where <math>A\subseteq\mathbb{R}</math>) is invertible on its range (image) if and only if it is either strictly [[monotonic function|increasing or decreasing]] (with no local [[maxima and minima|maxima or minima]]). For example, the function

: <math>f(x) = x^3 + x</math>

is invertible, since the [[derivative]]
{{math|1= ''f&prime;''(''x'') = 3''x''<sup>2</sup> + 1 }} is always positive.

If the function {{mvar|f}} is [[Differentiable function|differentiable]] on an interval {{mvar|I}} and {{math| ''f&prime;''(''x'') ≠ 0}} for each {{math|''x'' ∈ ''I''}}, then the inverse {{math|''f''<sup> −1</sup>}} is differentiable on {{math|''f''(''I'')}}.<ref>{{harvnb|Lay|2006|loc=p. 246, Theorem 26.10}}</ref>  If {{math|1= ''y'' = ''f''(''x'')}}, the derivative of the inverse is given by the inverse function theorem,

: <math>\left(f^{-1}\right)^\prime (y)  = \frac{1}{f'\left(x \right)}. </math>

Using [[Leibniz's notation]] the formula above can be written as

: <math>\frac{dx}{dy} = \frac{1}{dy / dx}. </math>

This result follows from the [[chain rule]] (see the article on [[inverse functions and differentiation]]).

The inverse function theorem can be generalized to functions of several variables. Specifically, a continuously differentiable [[real multivariable function|multivariable function]] {{math| ''f '': '''R'''<sup>''n''</sup> → '''R'''<sup>''n''</sup>}} is invertible in a neighborhood of a point {{mvar|p}} as long as the [[Jacobian matrix and determinant|Jacobian matrix]] of {{mvar|f}} at {{mvar|p}} is [[invertible matrix|invertible]]. In this case, the Jacobian of {{math|''f''<sup> −1</sup>}} at {{math|''f''(''p'')}} is the [[matrix inverse]] of the Jacobian of {{mvar|f}} at {{mvar|p}}.