Understanding the Chain Rule and Differentiability: Addressing Contradictions in Function Compositions

When discussing the chain rule in calculus and analysis, it is crucial to understand the logical structure and conditions under which it holds. In this article, we will explore a specific scenario where two function compositions, fgx and gfx, are examined for their differentiability at a point ( x 0 ), even though the function gx is not differentiable at that point. This will help clarify whether such a scenario contradicts the chain rule.

Introduction

The chain rule is a fundamental theorem in calculus and provides a method for finding the derivative of a composition of functions. It states that if g is differentiable at a point a and f is differentiable at ga, then the composition f°g is differentiable at a.

Functions and their Elements

Definitions of Functions

The functions in question are:

fx x2 gx x

Function Compositions

The compositions of these functions are:

fgx fx x2 gfx gx2 x2

Differentiability at ( x 0 )

Differentiability of ( fgx x^2 )

The function ( fgx x^2 ) is differentiable everywhere, including at ( x 0 ). At ( x 0 ), the derivative is:

[ (fgx)'left(0right) 2 cdot 0 0 ]

Differentiability of ( gfx x^2 )

The function ( gfx x^2 ) is also differentiable everywhere, including at ( x 0 ). At ( x 0 ), the derivative is:

[ (gfx)'left(0right) 2 cdot 0 0 ]

Differentiability of ( gx x )

The function ( gx x ) is not differentiable at ( x 0 ) because the left-hand and right-hand derivatives do not match:

( g_x^- -1 ) ( g_x^ 1 )

Chain Rule and the Given Scenario

The chain rule states that if ( g ) is differentiable at a point ( a ) and ( f ) is differentiable at ( ga ), then the composition ( fgx ) is differentiable at ( a ). In this case, ( gx x ) is not differentiable at ( x 0 ), but the composition ( fgx x^2 ) is differentiable because ( f ) is differentiable everywhere and the output of ( gx ) at ( x 0 ) is such that ( fg0 0 ). Therefore, the lack of differentiability in ( g ) at that point does not violate the chain rule since it does not require ( g ) to be differentiable at all points in its domain for the composition to be differentiable.

Matthew Smedberg correctly highlights that the chain rule has two key claims:

Existence Claim: If the derivative of f at a exists and the derivative of g at fa exists, then the derivative of fgx at a exists. Quantitative Connection: The derivative of fgx at a is given by the formula ( (fgx)'_a g'_f(a) cdot f'_a ).

The case when ( fa ) exists and ( gfa ) does not exist is not covered by the existence claim of the chain rule. This means that the composition can still be differentiable even if one of the functions involved is not differentiable at the specific point.

Conclusion

The fact that both ( fgx x^2 ) and ( gfx x^2 ) are differentiable at ( x 0 ) does not contradict the chain rule. The chain rule only requires the existence of the necessary derivatives for the composition to be differentiable, and it does not mandate that ( g ) be differentiable at every point throughout its domain.

Understanding these nuances helps in correctly applying the chain rule and interpreting its implications in various scenarios involving function compositions.