Difference between revisions of "Inverse function theorem"

From Calculus
Jump to: navigation, search
(Created page with "==Statement== ===Simple version at a specific point=== Suppose <math>f</math> is a function of one variable that is a one-one function and <math>a</math> is in the [[do...")
 
Line 1: Line 1:
 +
{{differentiation rule}}
 +
 
==Statement==
 
==Statement==
  

Revision as of 22:39, 21 September 2011

This article is about a differentiation rule, i.e., a rule for differentiating a function expressed in terms of other functions whose derivatives are known.
View other differentiation rules

Statement

Simple version at a specific point

Suppose f is a function of one variable that is a one-one function and a is in the domain of f. Suppose f is [differentiable function|differentiable]] at a and b = f(a). Suppose further that the derivative f'(a) is nonzero, i.e., f'(a) \ne 0. Then:

The inverse function f^{-1} is differentiable at b, and further:

(f^{-1})'(b) = \frac{1}{f'(a)}

Simple version at a generic point

Suppose f is a function of one variable that is a one-one function. Then, the formula for the derivative of the inverse function is as follows:

(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}

with the formula applicable at all points in the range of f for which f'(f^{-1}(x)) exists and is nonzero.

One-sided versions

Fill this in later

Infinity-sensitive versions

Fill this in later