Let f and g be the functions in Exercise 47 . (a) If F(x)=f(f(x)), find F^'(2). (b) If G(x)=g(g(

step 1 We have the function capital F of X, which is defined to be F of F of X, where lowercase F of X

Let f and g be the functions in Exercise 47 . (a) If...

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Key Concepts

Composite Functions

Composite functions are functions formed by applying one function to the output of another. In problems involving compositions like f(f(x)) or g(g(x)), understanding how to decompose the function into its constituent parts is critical. This understanding, combined with the chain rule, allows for the systematic calculation of derivatives.

Chain Rule

The chain rule is a fundamental principle in calculus for computing the derivative of a composite function. It states that if a function h(x) is composed as f(g(x)), then the derivative h'(x) is the product of f'(g(x)) and g'(x). This rule is essential when dealing with nested functions, such as f(f(x)) or g(g(x)).

Evaluating Derivatives at Specific Points

When a derivative is calculated through methods like the chain rule, it often needs to be evaluated at a particular point. This step involves substituting the appropriate value into the resulting derivative expression to determine the function's instantaneous rate of change at that point.

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