Use the given graphs of f and g to find each derivative. <IMAGE>
d/dx (f(f(x))) |x=4

Verified step by step guidance

1. Step 1: Understand the problem requires finding the derivative of a composite function, specifically f(f(x)), at x = 4.
2. Step 2: Apply the chain rule for derivatives, which states that the derivative of a composite function f(g(x)) is f'(g(x)) * g'(x).
3. Step 3: Identify the inner function g(x) as f(x) and the outer function f(g(x)) as f(f(x)).
4. Step 4: Evaluate the derivative of the outer function f'(f(x)) and the derivative of the inner function f'(x) at x = 4 using the given graphs.
5. Step 5: Multiply the derivatives from Step 4 according to the chain rule: f'(f(4)) * f'(4).

Key Concepts

Chain Rule

The Chain Rule is a fundamental principle in calculus used to differentiate composite functions. It states that if you have a function f(g(x)), the derivative is f'(g(x)) * g'(x). This rule is essential for finding the derivative of functions where one function is nested inside another, as in the case of f(f(x)).

Derivative

A derivative represents the rate of change of a function with respect to its variable. It is a measure of how a function's output value changes as its input value changes. Understanding how to compute derivatives is crucial for analyzing the behavior of functions, including finding slopes of tangent lines and optimizing functions.

Evaluating Functions

Evaluating functions involves substituting specific values into a function to determine its output. In this context, after finding the derivative of f(f(x)), you will need to evaluate it at x=4. This step is important for obtaining a numerical result that reflects the behavior of the composite function at that particular point.