Suppose that f(x)=x^2 and g(x)=|x| . Then the composites (f ∘g)(x)=|x|^2=x^2 and (g ∘f)(x)

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

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

Suppose that $f(x)=x^{2}$ and $g(x)=|x| .$ Then the composites
$$(f \circ g)(x)=|x|^{2}=x^{2} \quad \text { and } \quad(g \circ f)(x)=\left|x^{2}\right|=x^{2}$$
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

Composite Functions

Composite functions are formed by applying one function to the results of another. Even if one function in the composite is not differentiable at a specific point, the composition can still be differentiable if the non-differentiability does not 'transfer' to the composite. In other words, the structure of the composite may smooth out irregularities present in one of its parts, thereby sidestepping the prerequisites of the chain rule rather than contradicting it.

Differentiability

Differentiability refers to the existence of a derivative at a given point, meaning the function has a well-defined instantaneous rate of change there. A function may fail to be differentiable at certain points if, for example, there’s a sharp corner or cusp. However, differentiability of a composite function is not solely determined by the differentiability of its individual components at all points.

Chain Rule

The chain rule is a fundamental theorem in calculus that provides a method for differentiating composite functions. It states that if you have two functions that are both differentiable (with careful attention to the domains of differentiation), then the derivative of their composition is the product of the derivative of the outer function evaluated at the inner function and the derivative of the inner function. The key point is that the rule only applies when both functions are differentiable at the appropriate points.

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