Let f and g be the functions in Exercise 63 . (a) If F(x)=f(f(x)), find F^'(2) . (b) If G(x)=g(g

step 1 we have to use the same table which is given in question 63 to find the derivative of this rathe

Let f and g be the functions in Exercise 63 . (a) If...

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Key Concepts

Chain Rule

The chain rule is a fundamental concept in calculus used to differentiate composite functions. It states that if a function is composed of two functions, such as h(x) = f(g(x)), then the derivative is found by multiplying the derivative of the outer function evaluated at the inner function by the derivative of the inner function, i.e., h'(x) = f'(g(x)) * g'(x).

Function Composition

Function composition involves combining two or more functions into a single function, where the output of one function becomes the input of the next. This concept is essential in understanding operations such as F(x) = f(f(x)) or G(x) = g(g(x)), which require careful application of differentiation techniques.

Differentiation of Composite Functions

When differentiating a composite function like F(x) = f(f(x)), the derivative is found by applying the chain rule. Specifically, you differentiate the outer function f at f(x) and multiply by the derivative of the inner function f(x), resulting in F'(x) = f'(f(x)) * f'(x). This same approach applies to other composite functions as well.

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