Chain Rule Suppose that f(x)=x^2 and g(x)=|x| . Then the...

Chain Rule Suppose that $f(x)=x^{2}$ and $g(x)=|x| .$ Then the compositions$$(f \circ g)(x)=|x|^{2}=x^{2} \quad \text { and } \quad(g \circ f)(x)=\left|x^{2}\right|=x^{2}$$ are both differentiable at $x=0$ even though $g$ itself is not differentiable at $x=0 .$ Does this contradict the Chain Rule? Explain.

Chain Rule Suppose that $f(x)=x^{2}$ and $g(x)=|x| .$ Then the compositions$$(f \circ g)(x)=|x|^{2}=x^{2} \quad \text { and } \quad(g \circ f)(x)=\left|x^{2}\right|=x^{2}$$
are both differentiable at $x=0$ even though $g$ itself is not differentiable at $x=0 .$ Does this contradict the Chain Rule? Explain.

Key Concepts

Chain Rule

The chain rule is a fundamental principle in calculus that provides a method for finding the derivative of a composite function. Specifically, if you have a composite function h(x) = f(g(x)), and both f and g are differentiable at the relevant points, then the derivative h'(x) is given by f'(g(x)) multiplied by g'(x). This rule depends on the differentiability of each function to ensure a correct calculation of the derivative.

Differentiability

Differentiability refers to the property of a function having a derivative at a given point. A function that is differentiable at a point must behave smoothly enough there so that its instantaneous rate of change can be computed. It is possible, however, for a function to be continuous at a point without being differentiable, which introduces complexities when applying rules like the chain rule.

Composition of Functions

The composition of functions involves applying one function to the result of another, forming a new function. When composing functions, the overall smoothness or differentiability of the composite function can depend in subtle ways on the properties of the individual functions. In some cases, the combination can result in a function that is smoother than its components, even if one of them is not differentiable.

Cancellation of Non-Differentiability

In certain special cases, the structure of the composite function can cancel out the effects of non-differentiability present in one of its components. For example, if the non-differentiable behavior of an inner function does not affect the output of the outer function, the resulting composite function can be differentiable. This phenomenon does not contradict the chain rule because the chain rule assumes differentiability of the individual functions; when one function is not differentiable, the standard chain rule does not directly apply, but the resulting composition may nonetheless turn out to be differentiable due to algebraic cancellation.

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