Let f and g be the functions in Exercise 53 . (a) If...

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Key Concepts

Function Composition

Function composition involves applying one function to the result of another. When you have a composite function, such as f(f(x)) or g(g(x)), you first evaluate the inner function at x and then apply the outer function to that result. This concept is fundamental to understanding how nested functions interact.

Chain Rule

The chain rule is a technique in calculus used to find the derivative of a composite function. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. This rule is essential when differentiating functions like f(f(x)) or g(g(x)).

Derivative Evaluation

Evaluating the derivative at a specific point involves first finding the general expression for the derivative using appropriate rules (such as the chain rule) and then substituting the specific value into this derivative. This process allows you to determine the instantaneous rate of change of the function at that particular point.

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